scieee Science in your language
[en] (orig)
Nonlinear and Chaotic
Front Dynamics in
Semiconductor Superlattices
vorgelegt von
Diplom-Physiker
Andreas Amann
aus Rastatt
von der Fakult¨
at II – Mathematik und Naturwissenschaften
der Technischen Universit¨
at Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
– Dr. rer. nat. –
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. Erwin Sedlmayr
Berichter: Prof. Dr. Eckehard Sch¨
oll, PhD
Berichter: Prof. Dr. Harald Engel
Tag der wissenschaftlichen Aussprache: 11. Dezember 2003
Berlin 2004
D 83
Zusammenfassung
Die Ladungstr¨
agerdynamik in Halbleiter¨
ubergittern wird theoretisch auf der Grund-
lage eines semiklassischen sequentiellen Tunnelmodells untersucht. Abh¨
angig von
den Modellparametern ergeben sich dabei laufende oder station¨
are Anreicherungs-
und Verarmungsfronten der Elektronen.
Besonders interessante Szenarien entstehen dadurch, dass verschiedenartige Fron-
ten miteinander in Wechselwirkung treten und sich gegenseitig annihilieren k¨
onnen.
Durch diesen Mechanismus wird es m¨
oglich, ein chaotisches raum-zeitliches Verhal-
ten bei konstanter ¨
außerer Spannung herbeizuf¨
uhren. Das dabei auftretetende Bifur-
kationsverhalten wird analysiert. Es stellt sich heraus, dass ein ¨
aquivalentes Szenario
auch in einem System aus Tankbeh¨
altern auftritt, welche nach bestimmten Regeln
bef¨
ullt und entleert werden. Solche hybriden Tanksysteme werden ¨
ublicherweise bei
der Beschreibung von Lagerhaltungsproblemen in Fabriken eingesetzt. Ein Haupt-
schwerpunkt dieser Arbeit ist es, diese ¨
uberraschende Verbindung zweier vollst¨
andig
unterschiedlicher dynamischer Systeme zu begr¨
unden. Dazu wird zun¨
achst das Ver-
halten einzelner Fronten studiert, insbesondere deren Erzeugung am Emitterkon-
takt, sowie die Frontgeschwindigkeiten in Abh¨
angigkeit des durch das Bauteil flie-
ßenden Stromes. Anschließend wird das Zusammenspiel der Fronterzeugungs- und
Vernichtungsprozesse anhand einfacher Regeln erkl¨
art, welche durch weitere Spe-
zialisierung auf das erw¨
ahnte Tankmodell f¨
uhren. Im einfachsten Fall l¨
asst sich das
Tankmodell mittels einer eindimensionalen iterierten Abbildung analysieren, und es
werden die sich daraus ergebenden analytischen Ergebnisse mit den Resultaten aus
der numerischen Behandlung der vollen mikroskopischen Modellgleichungen vergli-
chen.
Weiterhin wird das dynamische Verhalten des ¨
Ubergitters bei nichtkonstanter
¨
außerer Spannung, wie zum Beispiel w¨
ahrend Schaltvorg¨
angen, betrachtet. Hier-
bei ergibt sich, dass das Schaltverhalten in nichttrivialer Weise von der Gr¨
oße der
Schaltspannung abh¨
angt. In diesem Zusammenhang wird auch das Verhalten unter
einer kombinierten Gleich- und Wechselspannung untersucht.
Durch eine Erweiterung der urspr¨
unglichen Modellgleichungen um eine zus¨
atzliche
Dimension senkrecht zur Haupttransportrichtung des Stroms wird es ferner m¨
oglich,
die Wechselwirkung lateraler und vertikaler Strukturen n¨
aher zu beleuchten.
iv
Abstract
The charge dynamics in semiconductor superlattices is studied theoretically on the
basis of a semiclassical sequential tunneling model. Depending on the model param-
eters, moving or stationary electron accumulation and depletion fronts are obtained.
Particularly interesting scenarios arise from the interaction between the fronts
and from the possibility of mutual front annihilation. With this mechanism it is
possible to induce chaotic spatio-temporal behavior at a fixed external voltage. By
analyzing the relevant bifurcations it turns out that an equivalent scenario also
occurs in a system of water tanks, which are filled and emptied following a given set
of rules. This type of hybrid tank system is known to be useful for the description
of stock-keeping problems in production systems. One main focus of this work is to
substantiate this surprising connection between two completely different dynamical
systems. For this purpose, we first study the dynamical behavior of single fronts, in
particular their generation at the emitter contact, as well as the front velocities as
a function of the overall current through the device. We then explain the interplay
of front generation and annihilation processes on the basis of simple rules, which
eventually lead to the tank model with further specialization. In the most simple
case, the tank model may be analyzed by means of a one-dimensional iterated
map, and we compare the analytical results with the numerical results from the full
microscopic model equations.
Also the dynamical behavior of the superlattice under nonstationary external
voltage conditions, such as during switching processes, is considered. It turns out
that the switching scenarios depend in a nontrivial way on the switching voltage.
In this context we also investigate the behavior at a combined ac and dc voltage.
By extending the original model equations with an additional dimension perpen-
dicular to the main current, we explore the interaction between lateral and vertical
structures.
vi
Contents
1 Introduction 1
2 The Microscopic Model 5
2.1 Vertical Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Quantum Transport . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Decoherence Theory . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The Sequential Tunneling Model . . . . . . . . . . . . . . . . . . . . 9
2.2.1 The Well to Well Characteristic . . . . . . . . . . . . . . . . 10
2.2.2 Global Coupling . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Boundary Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Front Dynamics in One Spatial Dimension 15
3.1 Dynamics of a Single Front . . . . . . . . . . . . . . . . . . . . . . . 15
3.1.1 The Current Velocity Characteristic . . . . . . . . . . . . . . 18
3.1.2 Depletion Front with Positive Velocity . . . . . . . . . . . . 22
3.1.3 Stationary Accumulation Front . . . . . . . . . . . . . . . . 23
3.1.4 Accumulation Front with Positive Velocity . . . . . . . . . . 24
3.1.5 Accumulation Front with Negative Velocity . . . . . . . . . 27
3.2 Multiple Fronts under Fixed External Voltage . . . . . . . . . . . . 28
3.3 Front Generation and Annihilation . . . . . . . . . . . . . . . . . . 30
3.3.1 Front Injection at the Emitter . . . . . . . . . . . . . . . . . 30
3.3.2 Front Collisions . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.3 Front Annihilation at the Collector . . . . . . . . . . . . . . 36
4 Chaotic Front Dynamics 37
4.1 Bifurcation Scenarios of the Microscopic Model . . . . . . . . . . . 37
4.1.1 The Case σ= 0.5 (Ωm)−1................... 37
4.1.2 Varying σ............................ 41
4.1.3 Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . 45
4.2 The Front model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2.1 Elimination of the current density . . . . . . . . . . . . . . . 47
4.2.2 The rules for the front model . . . . . . . . . . . . . . . . . 49
4.2.3 The case n=3 ......................... 52
4.2.4 Arbitrary n........................... 55
vii
Contents
5 The Tank Model 59
5.1 Deduction from the Front Model . . . . . . . . . . . . . . . . . . . . 59
5.1.1 The Case pl= 0 . . . . . . . . . . . . . . . . . . . . . . . . 59
5.1.2 The Case pl>0 . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Connection to Water Tanks . . . . . . . . . . . . . . . . . . . . . . 62
5.3 The Poincar´e Map . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.4 Bifurcation Analysis for n= 3 . . . . . . . . . . . . . . . . . . . . . 66
5.4.1 Connection with the Flat-Topped map . . . . . . . . . . . . 67
5.4.2 The Tent-Map Case λ= 1 . . . . . . . . . . . . . . . . . . . 69
5.4.3 The Case λ < 1 . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.4.4 Elementary Intervals . . . . . . . . . . . . . . . . . . . . . . 74
5.4.5 Period Doubling Cascade . . . . . . . . . . . . . . . . . . . . 75
5.4.6 Intermediate Intervals . . . . . . . . . . . . . . . . . . . . . 76
5.4.7 Symbolic Dynamics . . . . . . . . . . . . . . . . . . . . . . . 76
5.4.8 Chaoticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.5 Bifurcation Analysis for n= 4 . . . . . . . . . . . . . . . . . . . . . 78
6 Nonstationary External Voltage 81
6.1 Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.1.1 Down Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.1.2 Up Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2 Ramping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.3 Sweeping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.4 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7 Front Dynamics in Two Spatial Dimensions 97
7.1 Lateral Transport Theory . . . . . . . . . . . . . . . . . . . . . . . 97
7.1.1 Dynamical Equations . . . . . . . . . . . . . . . . . . . . . . 97
7.1.2 The Generalized Einstein Relation . . . . . . . . . . . . . . 98
7.1.3 Solving Poisson’s Equation . . . . . . . . . . . . . . . . . . . 98
7.2 Stability of Inhomogeneous Lateral Patterns . . . . . . . . . . . . . 100
8 Summary and Outlook 105
Bibliography 109
viii
List of Important Symbols
Symbol Description
U0/ U external voltage / voltage at superlattice
m∗
B/ m∗
Weffective electron mass in barrier / well
EB
c/ EW
cΓ point conduction band energy for barrier / well
Ψν
mk(z, r) Wannier wave function localized at well mwith lateral wave
vector kand band index ν
b / w barrier / well width
d / L period / total length of superlattice
Asample cross section
nmelectron density in well m
NDtwo-dimensional doping density
Fmelectric field between well mand well m+ 1
jm→m+1 current density from well mto well m+ 1
Nnumber of wells in superlattice
σOhmic boundary conductivity
jglobal current density
Fl(j)/ Fh(j) low and high field branch of electron density vs. electric field
characteristic
Fl
max / Fh
min largest / smallest field on low / high field branch
jl
max / jh
min largest / smallest current on low / high field branch
js
max / js
min largest / smallest current for which stationary fronts occur
Qa/ Qdcharge of a fully developed accumulation / depletion front
pa/ pdposition of accumulation / depletion front within superlattice
ai/ diposition of the ith accumulation / depletion front
va(j)/ vd(j) velocity of accumulation / depletion front
Na/ Ndnumber of accumulation / depletion fronts
jdcurrent density for which va(jd) = vd(jd)
j(Na,Nd)=jNa
Ndcurrent density for which Nava(j) = Ndvd(j)
(Fc, jc) intersection point of homogeneous current density characteris-
tic with emitter current density characteristic
Lhlength of high field region in superlattice
rcratio Na/Ndfor which j(rc) = jc
ix
Contents
Symbol Description
ph/ plminimum length of high / low field domain at emitter, before new
fronts can be injected
nmaximum number of fronts of one type (n= max [Nmax
a, Nmax
d])
xilength of the ith high field domain and filling height of tank #i.
µdraining rate for all tanks
λfilling rate of tank connected to server
Pn/ BnPoincar´e map / Poincar´e section of the n-tank model
MTII matrix for reordering of tank positions after switching
˜
Ndnumber of nonempty tanks before the next switching of the server
t−
m+1 / t+
m+1 time just before / after next switching
∆xemaximum amount of water lost by one tank before switching
z phin units of Lh, (z=ph/Lh)
ˆ
Pn
z(x)Pnrescaled in units of Lh
IP
zflat segment of ˆ
Pn
z
fλ(x) flat topped map on unit interval with maximum at λ= 1 −z/2
f1tent map
f(k)
λkth iterate of fλ
If
λflat segment of fλ
Qstring consisting of k−1 letters of 0 or 1
l= %Q Q is the binary representation of l
˜
Qbitwise inverse of Q
p(k)
l=pk
Qfixed points of f(k)
1on branches with positive slope
n(k)
l=nk
Qfixed points of f(k)
1on branches with negative slope
xi
λtrajectory of x0= 1/2 under fλ
k(λ) the shortest periodic orbits of fλhas the period k(λ)
Ik
QIk
Q=pk
Q, nk
Q
Ui/ Ufinitial / final voltage
Ustep Ustep =Uf−Ui
Ucrit tripole relocation triggered for Ustep > Ucrit
τrramping time
Udisc discontinuity in current voltage characteristic
τd/ τsdelay time / switching time
τrel relocation time τrel =τd+τs
Uac / Udc ac / dc part of the external voltage
jk
m→m+1 /j⊥
mtransversal / lateral current density
D0effective diffusion coefficient
x
1 Introduction
Moving fronts are the source of complex self-organized patterns in a broad range
of nonlinear systems. Starting from classical water waves, fronts appear in many
different forms in physics, such as the phase transition fronts in crystal growth
[1] or as interstellar conduction fronts [2] in astronomy. Prominent examples for
front systems in chemical science are the famous Belousov–Zhabotinskii reaction
[3, 4] or combustion waves [5]. Furthermore fronts are often a key element in the
self-organization processes in biological systems, for example the excitation wave in
cardiac tissue [6] or during morphogenesis [7]. It is therefore an important task of
nonlinear science to identify the basic features which are responsible for the similar-
ities and differences observed in a variety of front systems and to provide a unified
theory of front dynamics, which may explain the observed patterns irrespective of
the particular system at hand.
Since the 1960s many aspects of single isolated fronts have been studied in the
physical and mathematical literature. Thus a detailed understanding of the gen-
eration, the shape, and the propagation of single fronts in an infinite medium was
obtained in the context of simplified mathematical models in one or two dimensions
[8, 9]. In particular the importance of the non-equilibrium aspects was realized,
and important notions like the distinction between bistable, excitable and oscilla-
tory media were introduced.
In real world systems, however, multiple fronts often coexist, and the interaction
between fronts may lead to sophisticated self-organized patterns. To understand the
relevant mechanisms, it would be desirable to again obtain a simple mathematical
picture, which is capable of identifying the key elements that lead to a particular
pattern, but so far no unifying theory exists. Considerable effort in this direction
has been made concerning the problem of turbulence in fluid systems, which is often
quoted to be the “last great unsolved problem of classical physics” [10]. In spite of
major advances, a unified theory for turbulence is still not available and it is also
not clear, how the results in this area could be carried over to more general front
systems.
Semiconductor devices have a long tradition as practically relevant nonlinear
model systems [11, 12, 13, 14, 15, 16, 17, 18]. Fueled by their enormous technologi-
cal relevance and economic success, semiconductor materials have become one of the
best studied objects in solid state physics. The manufacturing technology for build-
ing small well defined semiconductor structures of high purity has steadily improved
during the last decades, and structures in the sub-micron range are commercially
available today.
1
1 Introduction
In a semiclassical description, the dynamically relevant quantities in semicon-
ductor devices are typically the densities of the free electrons or holes, the electric
field or the local temperature. Often the microscopic charge transport equations in
such devices are nonlinear [13, 18] and may give rise to a local region of Negative
Differential Conductance (NDC) in the local current density vs electric field char-
acteristic. For an S-shaped or Z-shaped local characteristic, the inhomogeneous
charge distribution is typically characterized by current filaments. Examples are
the Heterostructure Hot Electron Device (HHED) [19], thyristors [20] or the Double
Barrier Resonant Tunneling (DBRT) diode [21]. An N-shaped current density vs
electric field characteristic typically gives rise to charge accumulation and depletion
fronts forming electric field domains in the direction parallel to the current (vertical
fronts). Vertically moving charge fronts appear for instance in the Gunn diode [11].
In the following we will specifically consider semiconductor superlattices, which
consist of alternating layers of two different semiconductor materials. Using so-
phisticated growth techniques, such as the metal organic chemical vapor deposition
(MOCVD), it is possible to fabricate such alternating layers with a thickness of
only a few atomic monolayers at any desired doping density. What makes this
type of structure particularly interesting is the fact that they exhibit at the same
time, lateral filamentary structures and electron accumulation and depletion fronts
in the vertical direction. From the technological aspect, superlattices may serve as
a source for Gigahertz oscillations [22, 23, 24]. Recently, the successful operation
of a so called “quantum cascade laser” [25, 26], which is a specifically modified
superlattice, has sparked further interest in this type of structures.
It is the purpose of the present work to gain a better understanding of interact-
ing fronts, by using the semiconductor superlattice as a particularly simple, but
nevertheless technologically relevant model system. It will be shown that in this
case, a simple mathematical model can indeed satisfyingly predict the basic bifur-
cation scenarios. The methods which worked successfully in this case, could well be
generalized to suit other systems as well.
This thesis is organized as follows: After this introduction we will explain in
Chapter 2 the basic electron transport mechanisms leading to a sequential tunneling
model [27] for superlattices, which is used as the basis for the subsequent numerical
calculations. In the following Chapter 3 we study the generation and motion of
single fronts. It turns out that two complementary types of vertical fronts, namely
the electron accumulation front, and the electron depletion front exist. We examine
the velocity of single fronts as a function of the applied external current, and study
the motion of multiple fronts, which is governed by the global constraint of the
externally applied voltage. Particular consideration is given to the influence of the
contact boundaries on the generation and annihilation processes of new fronts.
In Chapter 4 we combine these results and derive a simplified front model, which
reproduces the numerically observed scenarios leading to chaos under a fixed exter-
nal voltage. Under the additional assumption that fronts do not traverse the whole
sample, we finally obtain in Chapter 5 a tank model, which explains the basic bi-
2
furcations by a set of filling rules for a system of water tanks. In the most simple
nontrivial case this system further reduces to a one dimensional map, which can be
analyzed analytically.
In the subsequent Chapter 6 we study the front dynamics under non-stationary
external voltage conditions, such as switching, ramping, or combined ac+dc volt-
age. Further interesting dynamical features occur if an additional lateral degree of
freedom for charge transport is taken into account. Such an extension is introduced
in Chapter 7, and the basic effects of interacting lateral and vertical fronts is ex-
amined. Finally in Chapter 8 we give a brief summary and an outlook for possible
directions of future research.
3
1 Introduction
4
2 The Microscopic Model
Grown-ups like numbers,
which make it unnecessary to
grasp the essential!
(Le Petite Prince)
In this chapter we will discuss the microscopic theory of the electron transport in
semiconductor superlattices. We will concentrate on the topics which are necessary
to understand the sequential tunneling model, which is used in the subsequent
chapters. For a broader coverage of the microscopic theory the reader is referred to
the books by Bastard [28] and Sch¨oll [29] as well as to the recent review articles by
Bonilla [30] and Wacker [27].
2.1 Vertical Transport
We consider a semiconductor superlattice, which consists of alternating layers of two
types of materials with different band gaps, such as AlAs and GaAs, or AlxGa1−xAs
and GaAs. We will only consider electron transport in n-doped samples. Then the
material with the lower conduction band edge will act as a quantum well, while the
other material represents a quantum barrier. The band structure for an AlAs/GaAs
superlattice is depicted in Fig. 2.1. The external voltage drop Uis applied in the
zdirection, i.e. perpendicularly to the quantum well layers, giving rise to a vertical
electron current.
2.1.1 Quantum Transport
Let us first consider a single electron in an infinitely long superlattice without bias.
The alternating layers of barrier and well material result in a z-dependence in the
potential energy of the electrons given by the conduction band edge Ec(z), and
also in the effective isotropic electron mass m∗(z).1The numerical values for the
1For the effective mass in the well material, we use the effective mass at the conduction band
edge mW
c. Since the electron energies we are interested in, can be located in the band gap of
the barrier material, it is not appropriate to use the effective mass mB
cat its conduction band
edge. Instead we use an energy dependent effective mass mB(E) in the barrier material as
proposed in [31], which interpolates linearly between zero at the valence band edge, and mB
c
at the conduction band edge.
5
2 The Microscopic Model
AlAs
GaAs
∆Ec
Conduction band
Ec(z)
Valence band
Ev(z)
E
z
Eb
Ea
d w b
∆b
∆a
EB
c
EW
c
EW
v
EB
v
EW
gEB
g
Figure 2.1: Schematic band structure of the conduction band Ec(z) and the valence
band Ev(z) in an AlAs/GaAs superlattice with barrier width b, well
width wand period d=w+b.EB
c/v and EW
c/v are the minimum energies
of the conduction band (c) and valence band (v) for the barrier (B) and
well (W) material, respectively; EB
gand EW
gare the respective energy
gaps between valence and conduction band. ∆Ec=EB
c−EW
cis the
difference in the conduction band energy of the well and the barrier
material. ∆aand ∆bdenote the widths of the first and the second
minibands, which are located at the energies Eaand Eb, respectively.
effective masses and relevant energies in the case of GaAs and AlAs are given in
Table 2.1.
Due to the periodicity of the structure, we have Ec(z+d) = Ec(z) and m∗(z+d) =
m∗(z), and obtain a Kronig-Penney type Hamiltonian [33],
H=−∇ ~2
2m∗(z)∇+Ec(z),(2.1.1)
with eigenfunctions of the form
ϕν
q,k(r, z) = eik·rϕν
q,k(z).(2.1.2)
GaAs AlAs
mc0.067 me0.15 me
Eg1.52 eV 3.13 eV
Ec0 1.05 eV
Table 2.1: Material parameters for GaAs and AlAs after [31, 32].
6
2.1 Vertical Transport
Here rand kare vectors in the two-dimensional (x, y) plane, νis the miniband index
and q∈[−π/d, π/d] is the quasi wave vector in z-direction. The Bloch functions
ϕν
q,k(z) are of the form
ϕν
q,k(z) = eiqzuν
q,k(z),(2.1.3)
with real dperiodic functions uν
q,k(z), and fulfill the one-dimensional Schr¨odinger
equation −∂
∂z
~2
2m∗(z)
∂
∂z +Vk(z)ϕν
q,k(z) = Eν
q,kϕν
q,k(z),(2.1.4)
with
Vk=Ec(z) + ~2k2
2m∗(z).(2.1.5)
Note that for all relevant energies m∗(zGaAs)< m∗(zAlAs). This leads to the
somewhat paradoxical situation that for large lateral momentum k, we obtain
Vk(zGaAs)> Vk(zAlAs), i.e. GaAs would act as a barrier, and AlAs attains the
role of the quantum well. This however is an artifact, which arises from the use
of the effective mass approximation at energies, at which it is not supposed to be
valid. In the following we will drop the kdependence of the Bloch functions by
setting ϕν
q=ϕν
qk=0.
A standard textbook solution of the Kronig-Penney model [28] yields the disper-
sion relation of the form
Eν
q,k=~2k2
2m∗(z)+Eν
q,(2.1.6)
where Eν
qis given by the implicit equation
cos(qd) = cos(kWw) cosh(κBb)−1
2m∗
BkW
m∗
WκB−m∗
WκB
m∗
BkWsin(kWw) sinh(κBb),
(2.1.7)
with kW=p2m∗
WEν
q/~and κB=q2m∗
B(∆Ec−Eν
q)/~.
2.1.2 Decoherence Theory
The Bloch functions ϕν
qdiagonalize the Hamiltonian Hin (2.1.6) for a single elec-
tron exactly, and appear therefore to be the most suitable basis from the quantum
mechanical point of view. However one apparent problem with the basis ϕν
qis that
they are completely delocalized. In practice however, the electrons appear to be
localized on a length scale of a few nanometers, and are not expected to extend
over the whole superlattice, which may be several hundreds of nanometers long.
This localization effect is even stronger, when inhomogeneous charge distributions
occur. Therefore we may ask, how the transition from the delocalized electron to a
localized one can be explained in a quantum mechanically correct way.
This question is related to the more general question, why objects appear to be
localized on a macroscopic scale, even if the quantum mechanical eigenstates are
7
2 The Microscopic Model
delocalized. This is answered conclusively by the so called decoherence theory which
was introduced by Zeh [34] in an attempt to overcome the measuring problem in
quantum mechanics. One of the basic observations in this theory is that due to
the continuous quantum mechanical interaction between all macroscopic objects, a
single object will not undergo the quantum mechanical transitions that would be
possible if the object was isolated [35, 36]. Instead, the objects tends to prefer the
basis, in which the interaction with the environment is diagonal. This so called
quantum Zeno effect has been verified in optical experiments [37]. The preference
of the position eigenstates, which results in the desired localization effect, is then
simply a consequence of the fact that all interactions are local, i.e. diagonal in
position space, but not in momentum space. This is in particular the case for the
Coulomb and the gravitational interactions, which are the dominant interactions
between macroscopic objects. These interactions therefore tend to produce many
particle entangled states |Ψiwhich are composed of components with approximate
eigenstates of the position operator, i.e.
|Ψi=X
i
ci|1ii|2ii···|nii,(2.1.8)
with ˆxn|nii ≈ xi|nii, where |niiis a single particle state. If there are many objects
present (n1), the phases between the individual components of the entangled
state become quickly randomized (hcicji= 0), which leads to decoherence, and
makes transitions between different components effectively impossible. On a small
scale with few particles however, the phases are not randomized, and quantum
coherence can be maintained for a long time.
The advantage of this approach is that there is no need for a singular measurement
process, which “collapses” the wave functions in the sense of the Copenhagen In-
terpretation of quantum mechanics. Instead, what is conceived as a measurement,
is merely the practical separation of different components of an entangled state,
and is completely explained within quantum mechanics. The question, whether an
interpretation of quantum mechanics along these lines necessarily leads to a “many-
worlds” interpretation [38] is still open to debate [35], and should not concern us
here. We conclude however that the decoherence theory can in principle quantita-
tively predict how the interaction with the environment influences the decoherence
length of the electrons. In particular, the intermediate case, where the electrons are
neither fully localized, nor fully delocalized could be calculated in a natural way.
The practical method to employ the decoherence theory, is to start with the
density matrix of the full state ρ=|ΨihΨ|which evolves according to the von Neu-
mann equation with the complete Hamiltonian. We may then concentrate on the
evolution of the first particle |1iby tracing out the other particles as environment
ρ1= tr2,...,nρ. (2.1.9)
The equation of motion for this subsystem can then be reduced to a Lindblad type
8
2.2 The Sequential Tunneling Model
equation [39]
i∂ρ1
∂t =H1, ρ1−iΛˆx1,ˆx1, ρ1,(2.1.10)
where H1is the reduced Hamiltonian and Λ summarizes the interaction of the
environment with the considered subsystem. Note that due to the non-unitary part
in (2.1.10), the subsystem can evolve from a pure state to a seemingly mixed state,
although the full density matrix ρremains in a pure state. The net effect of this
non-unitary part is that the non diagonal elements in the position basis of ρare
damped exponentially [35], i.e.
ρ(x, x0, t) = ρ(x, x0,0) exp −Λt(x−x0)2,(2.1.11)
which leads to a spatial decoherence.
In this work we are dealing with weakly coupled superlattices, where we may
assume that the spatial decoherence due to (2.1.11) is strong enough to localize
the electrons within one quantum well. It may be worthwhile, however to test the
implications of decoherence theory in the more critical case of strongly coupled
superlattices, where nontrivial predictions can be expected. It might for instance
be possible to strictly derive a continuum model of the type used in [40].
2.2 The Sequential Tunneling Model
For the sequential tunneling model we assume that the neighboring quantum wells
in the superlattice are weakly coupled, and therefore the localization effect due to
decoherence is large. It is then useful to work in a basis with localized wave functions
instead of energy eigenstates. A perfectly localized (delta peak) state would require
a superposition of all Bloch states in all minibands and is unphysical. But if we
restrict ourselves to superpositions of Bloch states from only one miniband we are
led to consider Wannier functions [41], defined by [42]
Ψν(z) = rd
2πZπ/d
−π/d
dqϕν
q(z).(2.2.1)
Here the phases of the different ϕν
qhave to be chosen in such a way that the best
localization in well number 0 is achieved. A general Wannier function localized at
well mand with lateral wave vector kis then given by
Ψν
mk(z, r) = Ψν(z−md)eik·r.(2.2.2)
Taking into account an additional electric field F, the Hamiltonian (2.1.1) can be
written in this new basis in a matrix of the form [27]
Hνµ
nk;mk0=(Eνδnmδνµ +
∞
X
h=1
(Tν
hδνµ −eFRνµ
h)δ(n+h)m+δ(n−h)m)δkk0.(2.2.3)
9
2 The Microscopic Model
Here the energy of the miniband Eν, and the coupling to the h-nearest neighbor
well Tν
his obtained from a Fourier expansion of the dispersion relation [42]
Eν
q=Eν+
∞
X
h=1
2Tν
hcos(hdq).(2.2.4)
In the following we will only take into account the coupling between neighboring
wells (h= 1). The electric field Fgives rise to an additional potential VF(z) = −eFz
yielding the matrix elements Rνµ
h=RdzΨν(z−hd)Ψµ(z).
If the decoherence time τφis small compared to the tunneling time, i.e. Γ =
~/τφTν
1√2, the phase information is lost between two tunneling events, and
the electron tunnels incoherently. Under this condition a sequential tunneling ap-
proach is justified [43]. If furthermore the thermodynamical relaxation between
the different energy levels of one quantum well is faster than the tunneling rate,
we can assume that each well is in a quasi-equilibrium state, characterized by a
quasi-electron temperature T. The effects due to electron heating and varying T
were studied in [44, 45], here we assume that the electron temperature Tcoincides
with the constant lattice temperature. If we furthermore assume that the lateral
sample cross section Ais so small that the electron concentration is homogeneous
in (x, y) direction, then the electron configuration of the superlattice is completely
determined by the electron densities nmin each quantum well m= 1 . . . N, where
Nis the number of quantum wells. Here nmis the number of electrons in well m
per sample cross section Aand is therefore a two-dimensional density. The electron
concentration changes according to the continuity equation
e˙nm=jm−1→m−jm→m+1,(2.2.5)
where e < 0 is the charge of the electron and jm→m+1 is the current density from
well mto well m+ 1.
2.2.1 The Well to Well Characteristic
Physically, the current density jm→m+1 should not only depend on the electron
densities nmand nm+1, but also on the electric field Fmbetween the two wells. The
electric field causes a relative shift of the miniband levels by Eν
m+eFmd=Eν
m+1. If
the miniband levels in two neighboring wells are aligned, i.e. Eν
m≈Eµ
m+1,resonant
tunneling without photon or phonon emission is possible. Since the scattering Γν
causes an energy broadening of the miniband states, a resonant current occurs even
if the level alignment is not perfect, but in general the current will decrease rapidly
with increasing level mismatch. We model this behavior by assuming that the
current density between the levels νand µis proportional to a Lorentzian of width
(Γν+ Γµ)/2. For kBTEb−Eawe may furthermore assume that the tunneling
current is dominated by electrons originating from one of the lowest energy levels
Ea
mor Ea
m+1, since at quasi-equilibrium this is the only significantly populated
10
2.2 The Sequential Tunneling Model
energy level. For the usual case F < 0 (electrons moving from left to right), we
have Eν
m> Eν
m+1, and the tunneling current from Ea
m+1 to Eν
mwith ν > 1 can be
neglected. jm→m+1 is then calculated by a Fermi’s Golden Rule like expression [42],
jm→m+1 =X
1ν
e
~Ha,ν
m,m+12Z∞
Ea
dEρ0nF(E−EF
m)−nF(E−EF
m+1 +eFd)
×Γ1+ Γν
(eFd +Ea−Eν)2+ (Γ1+ Γν)2/4,(2.2.6)
where nF(x) = (1+exp(x/kBT))−1is the Fermi function, EF
mis the Fermi energy in
well m,ρ0=m/π~2is the two dimensional density of state, and kBis Boltzmann’s
constant. Using
n=Z∞
Ea
dEρ0nF(E−EF
m) = ρ0kBTln 1 + exp EF
m
kBT,(2.2.7)
we may express the Fermi energy EF
mis a function of the electron density nm.
Performing the integration in (2.2.6) and replacing EF
mand EF
m+1 by nmand nm+1
via (2.2.7), we finally find
jm→m+1(Fm, nm, nm+1) = X
ν
e
~H1,ν
m,m+12
×Γ1+ Γν
(Eν−Ea−eFmd)2+Γa+Γν
22(2.2.8)
×nnm−ρ0kBTln he
nm+1
ρ0kBT−1e−eFmd
kBT+ 1io.
In the following we will only take into account the two lowest minibands. The
corresponding matrix elements can be obtained from (2.2.3) as Ha,a
m,m+1 =T1
1and
Ha,b
m,m+1 =−eFRa,b
1. Note that (2.2.8) is only valid for F < 0. For F > 0 we use
the identity
jm→m+1(Fm, nm, nm+1) = −jm→m+1(−Fm, nm+1, nm).(2.2.9)
For concreteness let us consider a superlattice of type A with the physical parame-
ters given in Table 2.2. The well to well characteristic jm→m+1(Fm, nm, nm+1) for dif-
ferent values nmis plotted in Fig. 2.2. The homogeneous characteristic nm=nm+1
(orange curve in Fig. 2.2) is point symmetric with respect to the origin as required
by (2.2.9), and shows four pronounced extrema which we label by the letters A, B,
C and D. The outer peaks A and D correspond to the resonant currents from Ea
mto
Eb
m+1 and Ea
m+1 to Eb
m, respectively. On the other hand, the two inner peaks B and
C are due to the tunneling current between Ea
m+1 and Ea
m. If we now keep nm+1
fixed at the doping density NDand vary nm, we find that peak D is not affected at
all, since the current from Ea
m+1 to Eb
monly depends on nm+1. On the other hand,
11
2 The Microscopic Model
Parameter Superlattice A Superlattice B
barrier AlAs Al0.3Ga0.7As
well GaAs GaAs
b[nm] 4.0 5.0
w[nm] 9.0 8.0
d[nm] 13.0 13.0
Ea[meV] 47.1 41.5
Eb[meV] 176.5 160
Γa[meV] 4.0 4.0
Γb[meV] 4.0 4.0
T[K] 5 20
Ra,b
1[mm] 0.268 12.7
N40 100
N3D
D[1015cm−3] 167 77
ND[µm−2] 2170 1000
Table 2.2: Superlattice parameters. Superlattices A corresponds to the experimen-
tal sample used in [46, 47]. Superlattice B is similar to the experimental
lattice in [48], but with a different doping density.
we find that the current density at peak A is proportional to nmand in particular
vanishes for nm= 0. The position of the peaks A and D does not change. This
is in contrast to peak C, which moves from Fm= 0 at nm= 0 to higher values
of |Fm|with increasing nm. At the same time the height of the peak decreases,
until it is hardly visible at nm= 5ND(Fig. 2.2). On the other hand peak B gets
more pronounced and moves towards Fm= 0 with increasing nm. We note that for
nm6=nm+1 the current density does not vanish at Fm= 0, since the Fermi energies
are different in both wells.
2.2.2 Global Coupling
The electron densities and the electric fields are coupled by the following discrete
version of Gauss’s law,
r0(Fm−Fm−1) = e(nm−ND) for m= 1, . . . N, (2.2.10)
where NDis the two–dimensional doping concentration, Nis the number of wells
in the superlattice, F0and FNare the fields at the emitter and collector barrier
and rand 0are the relative and absolute permittivities. Eq. (2.2.10) can be
derived from Gauss’s Law in the integral formulation, with the integration volume
being one well with finite width w. Then Eq. (2.2.10) follows under the conditions
that the charge is localized within the wells, and that the charge distribution does
not depend on the lateral coordinates. This is the case since the electron wave
12
2.3 Boundary Currents
-20-1001020 electric field [MV/m]
-0,2
-0,15
-0,1
-0,05
0
0,05
current density [A/mm2]
A
B
C
D
nm= 0
nm= 1 ND
nm= 3 ND
nm= 5 ND
Figure 2.2: Well to well characteristic jm→m+1(F, nm, nm+1) (eq. (2.2.8)) of super-
lattice A (Table 2.2), for nm+1 =NDand various values of nm.
functions are assumed to be Wannier functions in the vertical direction and plane
waves in lateral direction, and the background charge due to doping is assumed to
be confined to the center of the well.
The sum of the electric fields is then related to the total voltage drop Ubetween
emitter and collector by
U=−
N
X
m=0
Fmd. (2.2.11)
Here we choose a sign convention for the voltage, which makes Upositive, but the
electric fields and current densities are negative. The global coupling by the external
voltage U, will prove to decisively influence the dynamics of the superlattice. In
particular, Umay also depend on time as we will discuss in more detail in Chapter 6
(see also [49, 50]).
2.3 Boundary Currents
As will be discussed in Chapter 3, the proper choice of the boundary current j0→1
from the emitter to the first well decisively influences the dynamical front properties
of the superlattice.
For the following numerical calculations, we use the following simple Ohmic
13
2 The Microscopic Model
boundary current densities [44]:
j0→1=σF0,(2.3.1)
jN→N+1 =σFN
nN
ND
.(2.3.2)
where σis the Ohmic conductivity, and the factor nN/NDis introduced in order to
avoid negative electron densities at the collector. In Sec. 3.3.1 we will see that σ
governs the injection of electron accumulation and depletion fronts at the emitter.
In [51] a more microscopic method for calculating the current from a highly doped
emitter contact into the superlattice was proposed. A numerical implementation
[52] shows that such a scheme also generates electron accumulation and depletion
fronts at the emitter, which are equivalent to the fronts generated by the more
simple Ohmic boundary currents.
In [53] an exponential boundary current density of the form
j0→1=aexp (bF0),(2.3.3)
was considered. Here aand bare suitable parameters. Again it was found that the
dynamical behavior of the superlattice is equivalent to the dynamics under Ohmic
boundary currents.
From the experimental point of view, it is desirable to choose a specific dynamic
scenario by tuning the contact conductivity σ. Recent experimental studies show
that deep donors in the contact layers have a dramatic effect on the contact conduc-
tivity and a large increase of the contact resistance can be realized by decreasing
the temperature below 200 K [54]. Since this effect is sensitive to illumination, it
should be possible to adjust σoptically. Alternatively, the temperature dependence
of the emitter current may also be exploited.
Other boundary conditions (Dirichlet and Neumann) have been used in earlier
work [55, 56, 57, 58, 59, 60, 61].
14
3 Front Dynamics in One Spatial
Dimension
It is not necessary to
understand things in order to
argue about them.
(Pierre Augustin Caron de
Beaumarchais)
In a superlattice with a large number of quantum wells, charge accumulation and
depletion fronts typically occur, and play a major role in the dynamical behavior
of the system. Such fronts are either stationary or move with positive or negative
velocities. Two typical examples of front dynamics are shown in Fig. 3.1. We see
that particularly interesting scenarios may arise if fronts of opposite polarity collide
and annihilate (see Fig. 3.1(a)). In this chapter we will discuss the basic dynamics
of fronts in detail. The results of this chapter are the requisites for the front model
which will be introduced in Chapter 4.
Our analysis starts from the microscopic sequential tunneling model which was
explained in detail in Chapter 2. The model equations consist of the continuity
equation (2.2.5), the discrete version of Gauss’s law (2.2.10) and the global coupling
by the external voltage (2.2.11), which we compile here once more for convenience:
e˙nm=jm−1→m−jm→m+1 for m= 1, . . . N, (3.0.1)
r0(Fm−Fm−1) = e(nm−ND) for m= 1, . . . N, (3.0.2)
U=−
N
X
m=0
Fmd. (3.0.3)
3.1 Dynamics of a Single Front
The dynamics of single fronts in discrete systems have been extensively studied
in various contexts [62, 63, 64, 65], including the specific case of semiconductor
superlattices [66, 67, 68, 69, 27]. Although the general theory of front propagation
in discrete systems tends to become rather complicated [65], we will show that
the basic properties of fronts in semiconductor superlattices can be understood
easily by considering the “operating points” on the current density vs electric field
characteristic across each barrier.
15
3 Front Dynamics in One Spatial Dimension
well
1
100
well
1
100
well
1
100
well
1
100
a)
b)
Figure 3.1: Examples for the evolutions of the electron densities (top panels), elec-
tric fields (middle panels) and current densities (bottom panels) of su-
perlattice B for an external voltage U= 2 V and contact conductivity
σ= 0.5 Ω−1m−1(a) and σ= 1.3 Ω−1m−1(b). In the top panels the
electron accumulation and depletion layers are shaded in blue and red,
respectively. The red areas in the middle panels show the high field
domains. The raw current density data in the lower panels are plotted
in cyan, while the black lines show a running average of the current over
an interval of 0.5 ns.
16
3.1 Dynamics of a Single Front
Let us first consider the case of a single charge accumulation front, which is
located far away from the contacts. This front is characterized by a number of
consecutive quantum wells with indices ml,...,mr, where the electron densities
are noticeably larger than the doping density ND, whereas outside of the front the
electron densities are approximately equal to ND, i.e.
nm> ND+ 5% for m∈[ml, mr],(3.1.1)
nm=ND±5% else .(3.1.2)
Here a heuristic 5% accuracy cutoff is introduced since even far away from the front
the electron density is never exactly equal to the doping density. An analogous
definition for mland mrapplies in the case of a charge depletion front.
Instead of fixing the voltage drop Uat the device by (3.0.3), it turns out to be
advantageous to study the front motion at a fixed current density
j=1
N+ 1
N
X
m=0
jm→m+1 (3.1.3)
instead (here we neglected any contributions from the internal capacitance, since
we are interested in the current inside the sample). Practically this is achieved by
introducing a large external resistor R, and set
−U=−U0−RAj, (3.1.4)
where Ais the sample cross section, and U0is the fixed overall voltage. For a
sufficiently large Rwe have |U| |RAj|. The current density is then approximately
fixed by
−j=U0
RA −U
RA ≈U0
RA.(3.1.5)
Note that Uitself is not assumed to be fixed. However, a change in Udue to the
internal degrees of freedom of the superlattice will only have a tiny effect on j,
according to (3.1.5).
A typical profile for the electron density and the electric field of an electron
accumulation front under fixed current density conditions is shown in Fig. 3.2. In
this case the front width is about 6 wells.
Far away from the front, the well-to-well current densities obey the homogeneous
current density vs field characteristic as in Fig. 3.3. Furthermore the electric field
must be located on one of the branches with positive differential conductivity, since
otherwise the configuration would not be stable against small charge fluctuations.
For a fixed current density, this determines the low and high fields Fl(j) and Fh(j),
respectively (see Fig 3.3). The field obeys Gauss’s law (3.0.2) and therefore in-
creases1from Fl≈0 to a large negative value Fhwith increasing well number m.
1Due to the negative sign of the electron, the electric fields and current densities are negative
for our choice of the coordinate system. It is nevertheless customary to call Fhthe high field
and Flthe low field, although formally 0 > F l> F h. Consequently, terms like increasing and
decreasing are used in reversed logic in connection with fields and current densities.
17
3 Front Dynamics in One Spatial Dimension
-6
-4
-2
0
2
electric field [MV/m]
40 45 50 55 60
well index m
-2
0
2
4
electron density (nm-ND) [ND]
Figure 3.2: Electron density (black) and electric field (red) profile for a stationary
charge accumulation front at constant current density. j=−6.0 A/mm2
The total charge Qa<0 per unit area in the accumulation front is then simply
given by
Qa(j) =
mr
X
m=ml
e(nm−ND) = r0(Fh(j)−Fl(j)).(3.1.6)
Here we assume that the current is fixed to the same value at both sides of the
front, which is only possible if the current density is chosen in the interval where
the multistability in the field occurs (cf. Fig. 3.3). Otherwise Qawould be time-
dependent, and the front would be unstable.
In the case of an electron depletion front, the electron density and field profiles
are shown in Fig. 3.4. The electric field shows a drop from Fh(j) to Fl(j) with
increasing well index m. By comparing with (3.1.6) it is obvious that the total
charge of the depletion front is Qd=−Qa. We furthermore note that the charge
profile of the depletion front is flatter and broader than for the accumulation front.
The reason for this difference is that the electron density nmis required to be
positive. Therefore the contribution of one well to the total charge Qdcan not
exceed −eND. Such a restriction does not apply for charge accumulation fronts,
since there is no upper limit on nm. In fact we see from Fig. 3.2 that for this choice
of parameters, the majority of the charge in an accumulation front is located within
one single well.
3.1.1 The Current Velocity Characteristic
In order to study the motion of charge fronts it is useful to define the position pa/d
of the electron accumulation or depletion front by its center of charge,
pa/d =
mr
X
m=ml
mde(nm−ND)
Qa/d
.(3.1.7)
18
3.1 Dynamics of a Single Front
-10
-5
0
5electric field [MV/m]
-30
-20
-10
0
10
20
30
current density [A/mm2]
Fh
Fl
jh
min
jl
max
Fh
min
Fl
max
Figure 3.3: Homogeneous well-to-well current density vs field characteristic for a su-
perlattice of type B. Fland Fhdenote the low and high field region on
the first and third branch of the characteristic, respectively. The transi-
tion from the first branch to the second branch occurs at (Fl
max, jl
max) =
(−0.36 MV/m,−17.6 A/mm2) and the transition from the second to the
third branch at (Fh
min, jh
min) = (−3.95 MV/m,−1.10 A/mm2). Only the
second branch exhibits negative differential conductivity. The orange
double headed arrows indicate the possible ranges for Fland Fh.
Note that pa/d is a real number, although the underlying superlattice is discrete.
The velocity va/d of an accumulation or depletion front can then be obtained by
differentiating (3.1.7) with respect to time and using the continuity equation (3.0.1)
va/d = ˙pa/d =
mr
X
m=ml
mdjm−1→m−jm→m+1
Qa/d
(3.1.8)
=d
Qa/d mljml−1→ml+
mr−1
X
m=ml
jm→m+1 −mrjmr→mr+1!(3.1.9)
≈d
Qa/d
mr−1
X
m=ml
(jm→m+1 −j),(3.1.10)
19
3 Front Dynamics in One Spatial Dimension
-6
-4
-2
0
2
4
6
electric field [MV/m]
50 55 60 65 70
well index m
-1
-0.5
0
0.5
1
electron density (nm-ND) [ND]
Figure 3.4: Electron density (black) and electric field (red) profile for a charge de-
pletion front moving with positive velocity at a constant current density
j=−2.0 A/mm2.
where in the last step we have used that jml−1→ml≈jmr→mr+1 ≈j, which is fulfilled
to a high degree of accuracy for all current densities outside the front as defined by
(3.1.1).
Further insight into the term jm→m+1 −jappearing in (3.1.10) can be gained
by differentiating Gauss’s law (3.0.2) with respect to tand using the continuity
equation (3.0.1) to arrive at
r0
dFm
dt=j−jm→m+1 for m= 0 . . . N. (3.1.11)
Using (3.1.11) together with (3.0.2) and (3.0.3) leads to an alternative set of dy-
namical model equations in terms of electric fields, instead of electron densities,
which is well studied in the literature [70, 71].
Substituting (3.1.11) into (3.1.10) and using the fact that ˙
Fm= 0 for m /∈[ml, mr]
we obtain
va/d =−d
Qa/d
N
X
m=0
r0
dFm
dt.(3.1.12)
Using (3.0.3) and (3.1.6) finally yields the simple relation
va/d =±1
Fh(j)−Fl(j)
dU
dt.(3.1.13)
We may use (3.1.13) to obtain the front velocities as a function of jnumerically.
For this purpose, we approximately fix the current density jusing a large load
resistor (RA = 109Ωm2) according to (3.1.5). We then calculate the slope of the
sample voltage U(t) by numerical regression. The corresponding results are shown
in Fig. 3.5. For the depletion front (red line in Fig. 3.5) we obtain an always
20
3.1 Dynamics of a Single Front
0-5 -10 -15
current density [A/mm2]
60
40
20
0
-20
-40
velocity [wells/ns]
js
max
js
min
j d / (eND)
acc. front: theory
depletion front
accumulation front
unstable fronts
Figure 3.5: Front velocity vs current density for electron accumulation (blue) and
depletion (red) fronts of superlattice B. The broken lines denote unstable
fronts. js
min and js
max denote the minimum and maximum current for the
stationary accumulation front. The orange and green lines are analytical
predictions of the accumulation and depletion front velocities according
to (3.1.25) and (3.1.14), respectively.
positive velocity which is approximately proportional to the current density. For
small current densities however the depletion front becomes unstable (broken line)
which is due to the fact that the high field branch Fh(j) of the homogeneous current
density characteristic can not support arbitrarily small currents, but has a minimum
at jh
min ≈ −1.15 A/mm2(see Fig. 3.3). If we try to impose an external current
density below jh
min, this will only affect the low field region, which is at the right of
the front. Consequently more electrons are entering the front from the left than are
leaving at the right border, until the depletion front has vanished.
For the electron accumulation front (blue line in Fig. 3.5) the velocity vs current
density characteristic is more complicated. For small currents the front is unstable
for the same reasons as the depletion front above. With increasing current the
velocity drops from positive values to zero, which means that the front becomes
stationary. The fact that the front can be pinned for a finite range of jis due
to the discreteness of our system, and would disappear in the continuous limit
N→ ∞,d→0. With further increase of the current, the front is unpinned
and starts to move with negative velocity, i.e upstream towards the emitter [67].
Since currents larger than jl
max ≈ −17.5 A/mm are not supported by the low field
branch of the homogeneous characteristic (Fig. 3.3) the accumulation fronts become
unstable beyond jl
max.
21
3 Front Dynamics in One Spatial Dimension
-10
-5
0electric field [MV/m]
-20
-15
-10
-5
0
current density [A/mm2]
well 55 to 56
well 56 to 57
well 57 to 58
well 58 to 59
well 59 to 60
well 60 to 61
well 61 to 62
j = -2 A/mm2
Figure 3.6: Various well-to-well characteristics jm→m+1(F, nm, nm+1) for the charge
depletion front of Fig. 3.4 at j=−2 A/mm2. The framed squares
denote the actual operating points (Fm, jm→m+1(Fm, nm, nm+1)) if the
field profile of Fig. 3.4 is taken into account.
3.1.2 Depletion Front with Positive Velocity
The simplest case of front propagation is that of a depletion front (Fig. 3.4). For m∈
[ml, mr] the current density vs electric field characteristic jm→m+1(F, nm, nm+1) will
not simply obey the homogeneous characteristic of Fig. 3.3, since nmand nm+1 are
different from ND. But if the electron density profile nmis known for one particular
front, we can calculate the inhomogeneous characteristic jm→m+1(F, nm, nm+1) as a
function of Fat each mseparately.
The resulting current density characteristics are shown in Fig. 3.6. We see that at
the left and right borders of the front we obtain an almost homogeneous character-
istic (black and yellow line in Fig. 3.6), since there the electron densities are not too
different from the doping density. Inside the front the electron concentration is de-
pleted and almost vanishes at the center of the front (see well 58 and 59 in Fig. 3.4).
Following the discussion in Section 2.2.1, this results in severely suppressed current
density characteristics (red and magenta lines in Fig. 3.6), which are in particular
below the external current j(green line) for any field between Fland Fh.
Let us now consider the operating points (Fm, jm→m+1). At the left boundary of
the front the operating point is close to (Fh(j), j) (black square in Fig. 3.6). With
increasing well index mthe field Fmdecreases towards Fland the current jm→m+1
drops to almost zero and rises again to j. We therefore note that all contributions
to the velocity in (3.1.10) are positive, and we conclude vd>0.
A useful approximation for the velocity vdcan be obtained by considering (3.1.11)
at the center of the front, where we can approximate nm≈0. Then we have
jm→m+1 = 0, and ˙
Fm=j/(r0). From Gauss’s law (3.0.2) we know on the other
22
3.1 Dynamics of a Single Front
-10
-5
0electric field [MV/m]
-20
-15
-10
-5
0
current density [A/mm2]
well 49 to 50
well 50 to 51
well 51 to 52
well 52 to 53
j = -6 A/mm2
Figure 3.7: Well-to-well characteristics as in Fig. 3.6, but for a stationary charge ac-
cumulation front at j=−6.0 A/mm2. For the electric field and electron
density profile of this front see Fig. 3.2.
hand that Fm−1=Fm+eND/(r0). The time ∆tat which Fm(t+ ∆t) = Fm−1(t)
is then given by ∆t=eND/j. But ∆tis also the time needed for the front to travel
by one well period d. Thus the velocity of the depletion front is positive and can
be approximated by [68, 66]
vd≈jd
eND
.(3.1.14)
From Fig. 3.5 we see that this approximation is in very good agreement with the
numerical calculations, except for current densities close to the front instability.
However, we stress that (3.1.14) is only valid for rather low doping density, i.e.
−eND< Qdsince the derivation depends on the presence of at least one completely
depleted well with nm≈0. It was in fact shown by Wacker [27] that for high doping
values even negative velocities for vdare possible.
3.1.3 Stationary Accumulation Front
Let us now consider a stationary accumulation front, i.e ∂tnm= 0 for all mat a
fixed external current density j(cf. Fig. 3.2). Let us denote by mpthe well with
the highest electron concentration (for Fig. 3.2 we have mp= 51). The individual
well-to-well characteristics close to mpare shown in Fig. 3.7.
We see that by approaching the front from the emitter side, we first observe
an almost homogeneous characteristic (blue line in Fig. 3.7) and a current density
electric field operating point close to (Fl, j) (blue square). But already at the
next barrier the current density characteristic jmp−1→mp(F) (cyan line) shows a
suppressed low field peak. This is due to the large electron concentration at well
mpinhibiting the tunneling of electrons into well mp, as discussed in Sec. 2.2.1. Since
23
3 Front Dynamics in One Spatial Dimension
-6
-4
-2
0
2
electric field [MV/m]
45 50 55 60 65
well index m
-2
0
2
4
electron density (nm-ND) [ND]
Figure 3.8: Electron density (black) and electric field (red) profile for a charge
accumulation front moving in positive direction at constant current
j=−2.0 A/mm2.
the electric fields are constant in time, we conclude from (3.1.11) that in particular
jmp−1→mp=j, while the electric field Fmp−1is larger than Fl(cyan square). At the
next barrier the current density takes advantage of the large electron density nmp
which yields a characteristic jmp→mp+1(F) (magenta line). The electric field Fmp
(magenta square) has increased by a large amount due to nmp> ND, but the current
is fixed at jmp→mp+1 =j. For even larger mthe characteristic again approaches the
homogeneous characteristic and the operating point is close to (Fh, j). Since at any
barrier we have jmp→mp+1 =j, the total velocity of the front is zero according to
(3.1.10). Note that none of the operating points is located at the unstable branch
with negative differential conductivity. This is only possible in a discrete system,
where the field changes by a finite amount from one barrier to the next. It thus
follows, that stationary fronts of this type can not appear in a continuous system,
since there the branch with negative differential conductivity can not be avoided.
3.1.4 Accumulation Front with Positive Velocity
By lowering the external current jwe arrive at well-to-well characteristics as in
Fig. 3.9. By comparing with Fig. 3.7 we note that the characteristics themselves
did not change considerably, but only the imposed external current j(green line) is
lowered. In particular there is now no operating point, at which the characteristic
jmp→mp+1(F) could assume j, i.e the magenta characteristic and the green line in
Fig. 3.9 do not intersect. Instead we have jmp→mp+1(Fmp)< j, which results in a
positive velocity by (3.1.10).
In order to estimate the velocity of the accumulation front with positive velocity it
is instructive to look once more at the stationary case, just before the front starts to
move. The corresponding characteristics for the minimal stationary current js
min =
24
3.1 Dynamics of a Single Front
-10
-5
0electric field [MV/m]
-20
-15
-10
-5
0
current density [A/mm2]
well 53 to 54
well 54 to 55
well 55 to 56
well 56 to 57
j = -2 A/mm2
Figure 3.9: Well to well characteristics for a right moving charge accumulation front
at j=−2.0 A/mm2(profile in Fig. 3.8).
-10
-5
0electric field [MV/m]
-20
-15
-10
-5
0
current density [A/mm2]
well 51 to 52
well 52 to 53
well 53 to 54
well 54 to 55
j = -3.7 A/mm2
Figure 3.10: Well to well characteristics for a stationary charge accumulation front
with j=−3.7 A/mm2close to js
min.
25
3 Front Dynamics in One Spatial Dimension
−3.7A/mm2is shown in Fig. 3.10. We see that now the external current density j
(green line) intersects jmp→mp+1(F) at its minimum value. We may approximate the
homogeneous current density vs electric field characteristic jm→m+1(Fm, ND, ND) for
Fh(j)< F < F l
max by a parabola going through (Fl
max, jl
max) and with a minimum
at Fh
min. For the inhomogeneous case we can assume that
jm→m+1(Fm, nm, nm+1)≈(nm/ND)jm→m+1(Fm, ND, ND),(3.1.15)
which is appropriate if Pauli blocking can be neglected. This yields (in the fully
degenerate limit)
jm→m+1(Fm, nm, nm+ 1) ≈nm
ND"jh
min +jl
max −jh
minFm−Fh
min
Fl
max −Fh
min 2#.(3.1.16)
We can determine the minimal stationary current js
min by requiring that the region
with negative differential conductivity (NDC) is traversed in one step, i.e. Fmp−1≈
Fl
max and Fmp≈Fh
min (see Fig. 3.10). Then we have
nmp(Fmp)≈ND+r0
eFmp−Fl
max,(3.1.17)
and arrive at the well know condition for stationarity [42, 72] (see also [71, 59])
js
min ≈jmp→mp+1(Fh
min, nmp, nmp+ 1) (3.1.18)
≈1 + r0
eNDFh
min −Fl
maxjh
min ≈ −3.99 A/mm2.(3.1.19)
which overestimates the numerical value js
min =−3.64 A/mm2of superlattice B
slightly by about 10%.
By lowering jbelow js
min the field Fmpwill move according to (3.1.11). We can
estimate the time ∆tduring which Fmdrops from Fmpto Fmp−1, which equals the
time in which the front advances by one well. From (3.1.11) we obtain
∆t=ZFmp−1
Fmp
r0
j−jmp→mp+1(Fm, nm, nm+1)dFm.(3.1.20)
We can assume that the main contribution to the integral in (3.1.20) arises from
the region close to Fm≈Fh
min, where the integrand has a maximum, while the
exact choice of the boundaries is not crucial. Using the approximations (3.1.16)
and (3.1.17) and the substitution
x=Fmp−Fh
min
Fl
max −Fh
min
,(3.1.21)
we get
∆t≈ −αeND
jh
min Z1
−1
dx
j
jh
min −(1 + α) + αx −βx2+αβx3.(3.1.22)
26
3.1 Dynamics of a Single Front
-8
-6
-4
-2
0
2
electric field [MV/m]
40 45 50 55 60
well index m
0
2
4
electron density (nm-ND) [ND]
Figure 3.11: Electron density (black) and electric field (red) profile for a charge
accumulation front moving in negative direction at constant current
j=−10.0 A/mm2.
Here we used the abbreviations
α=js
min −jh
min
jh
min
=r0
eNDFh
min −Fl
max,(3.1.23)
β=jl
max −jh
min
jh
min
,(3.1.24)
and for convenience integrate over the interval [−1,1]. The numerical integration of
(3.1.23) for given αand βis straightforward. For the parameters of superlattice B
we have α≈2.63 and β≈15.0. The velocity vais then obtained by
va=d
∆t,(3.1.25)
and is plotted in Fig. 3.5 (orange line). In spite of the coarse approximations made,
the analytical approximation agrees astonishingly well with the results from the
numerical simulations. In particular the crossing point of the velocities of the accu-
mulation and depletion fronts is well reproduced by the analytical approximations
(3.1.25) and (3.1.14). At low values of j≈jh
min however, where the accumula-
tion front becomes unstable, the velocity obtained from (3.1.25) underestimates va
considerably.
3.1.5 Accumulation Front with Negative Velocity
Besides positive and zero velocities, the electron accumulation fronts show negative
velocities for external currents larger than js
max (see Fig. 3.5) [67]. For a charge
accumulation front moving left, the charge and field profiles (Fig. 3.11) are very
27
3 Front Dynamics in One Spatial Dimension
-10
-5
0electric field [MV/m]
-20
-15
-10
-5
0
current density [A/mm2]
well 45 to 46
well 46 to 47
well 47 to 48
well 48 to 49
j = -10 A/mm2
Figure 3.12: Well to well characteristics for the left moving charge accumulation
front in Fig. 3.11.
similar to the stationary case (Fig. 3.2). Consequently the well to well characteristics
in Fig. 3.12 are also similar to the stationary ones (Fig. 3.7) but with an external
current j(horizontal green line in Fig. 3.12) at a higher value. Due to this rise
of jthere is now no intersection point of the characteristic jmp−1→mp(F) (cyan
line) with j(green line) on the first branch. This means that the operating point
(jmp−1→mp, Fmp−1) (cyan square) is below jand results in a negative contribution
in (3.1.10). Since all other operating points are also less than or equal too j, we
conclude that vdin this regime will be negative.
In principle, it is possible to carry out a similar analysis for the negative front
velocity as was done for the positive front velocities in Sec. 3.1.4. It is clear that
the main contributions to an integral corresponding to (3.1.20) will now arise from
the region F≈Fl
max, which is responsible for the moving instability. However in
this case the approximation of the current density vs field characteristic close to
Fl
max is more complicated, since now the dependence of jm→m+1 on nm+1 can not
be neglected. In particular the approximation (3.1.16) is not sufficient for negative
velocities, and a diffusion term has to be included in the analysis. Since in this
work we are mainly concerned with currents below js
min we do not pursue this path
any further, but refer the reader to [27], where negative velocities were analyzed in
a continuous limit model.
3.2 Multiple Fronts under Fixed External Voltage
In the previous section we have studied the motion of a single front at fixed external
current jfar away from the contact and obtained the front velocity vs current
density characteristic in Fig. 3.5. We now consider the case of several fronts, which
28
3.2 Multiple Fronts under Fixed External Voltage
0 -1 -2 -3 -4
current density [A/mm2]
30
25
20
15
10
5
0
velocity [wells/ns]
j(3,4) j(4,3) j(2,1)
j(1,2)
j(2,3) j(3,2)
jd
depletion front
accumulation front
Figure 3.13: Velocity vs current density characteristic as in Fig. 3.5, in the region
around jd, where the accumulation and depletion front velocities are
equal. j(Na,Nd)denote the points where Nava=Ndvd(cf. (3.2.2)).
respectively.
are assumed to be well separated and far away from the contacts. Instead of fixing
jwe now fix the external voltage U, which is experimentally much more convenient.
Since the fronts are assumed to be separated, the indices mland mrin (3.1.7) are
well defined for each individual front, and we may therefore calculate the positions of
each accumulation front a1. . . aNaand depletion front d1. . . dNd. Here Naand Ndare
the number of accumulation and depletion fronts, respectively. Since accumulation
and depletion fronts appear alternatingly in the vertical direction, we have
Na−Nd= +1,0,−1.(3.2.1)
We may now conceive the superlattice as being split into Na+Ndsmaller parts,
each of which contains only one front. Since the total charge in each part is fixed to
either Qaor Qdthe current density at the boundaries of each part is also fixed to
the same value jthroughout the superlattice. We may therefore apply the results
of Section 3.1 to each part separately. By summing (3.1.13) over all parts, and
assuming that the external voltage Uis constant we get the important relation:
Nava(j) = Ndvd(j).(3.2.2)
Note that this relation is exact in the limit of well separated fronts.
If the number of fronts Naand Ndis given, the current density jis fixed by
(3.2.2). In the case Na=Nd, i.e. an even number of fronts, we have j=jd, where
jdis at the intersection point of va(j) and vd(j), see Fig. 3.13. Similarly for the
tripole configuration consisting of two accumulation fronts and one depletion front,
29
3 Front Dynamics in One Spatial Dimension
the current density j=j(2,1) is fixed by 2va(jt) = vd(jt). For other configurations
the corresponding current densities are described in Fig. 3.13. Since there is only
a countable set of configurations, the set of possible jis discrete with jdbeing the
only limit point.
With this knowledge we can now explain the current density trace of Fig. 3.1(b),
which alternates between a dipole and a tripole configuration. For a dipole configu-
ration with one accumulation and one depletion front, the averaged current density
is fixed to the constant value jd, while in the tripole configuration with two deple-
tion and one accumulation fronts, we obtain j=j(2,1) as predicted from (3.2.2).
In this context it is instructive to realize the meaning of (3.2.2) directly from the
field evolution (middle panel of Fig. 3.1(b)) in the tripole phase. Due to the fixed
voltage, the total length of the red high field domain is required to be constant.
Since the high field domain shrinks with the motion of the two accumulation fronts
and increases by the depletion front, the velocity of the depletion front obviously
has to be twice the velocity of the accumulation fronts.
In contrast to the averaged current density, the raw current density data (cyan line
in Fig.3.1(b)) shows rapid spikes which are due to the discreteness of the superlattice
[52] (well-to-well hopping of charge packets, cf. [73]).
In the current density trace of Fig. 3.1(a) we also observe plateaus corresponding
to the currents density j(1,2),j(2,3),j(3,4) and jd, although they are not as flat and
well developed as in Fig. 3.1(b). The reason for this difference will become clearer
in Chapter 4.
3.3 Front Generation and Annihilation
So far we have only considered the free motion of charge fronts well separated from
each other and the contacts. But for interesting dynamical scenarios as for example
in Fig. 3.1(a), we also need front generation and front annihilation processes.
3.3.1 Front Injection at the Emitter
In the superlattices under consideration, both types of fronts are in general only
generated at the emitter. We will see that the choice of the boundary conditions, as
well as the imposed external current jplay a decisive role for the front generation.
For convenience we assume that the emitter contact is Ohmic (2.3.1),
j0→1(F0) = σF0,(3.3.1)
with σthe contact conductivity, and F0the electric field between the emitter and the
first well. In the following we will choose σsuch that the linear contact characteristic
j0→1(F0) intersects the N-shaped homogeneous characteristic j1→2(F1, ND, ND) at
a point (Fc, jc) on its branch with negative differential conductivity (see Fig. 3.14).
30
3.3 Front Generation and Annihilation
-8-6-4-20 electric field [MV/m]
-20
-10
0
current density [A/mm2]
(Fc, jc)
σ = 1.0 Ω−1m-1
j = -1.5 A/mm2
j = -3 A/mm2
Figure 3.14: Emitter current density characteristic compared to homogeneous well
to well characteristic jm→m+1(F, ND, ND). The intersection point of
the two characteristics is denoted by (Fc, jc).
0 10 20 30 40 50
σ [Ω−1m-1]
0
-5
-10
-15
-20
current density [A/mm2]
no front generated
depletion front
accumulation front
jc(σ)
Figure 3.15: Intersection point of the homogeneous characteristic with the emitter
characteristic jc(σ). The red (blue) squares denote a successful gen-
eration of a depletion (accumulation) front at the emitter. The green
squares denote that no moving front was generated.
31
3 Front Dynamics in One Spatial Dimension
Let us consider a superlattice under fixed external current density j, with initial
conditions
ni(t= 0) = ND;F0(t= 0) = j/σ. (3.3.2)
It follows from (3.1.11) that F0is a stable fixed point. From (3.0.2) we see that
F1(t= 0) = F0. Let us first assume that jis larger than jc(red horizontal line
in Fig. 3.14). In this case F0and F1(t= 0) are larger than Fc(red squares in
Fig. 3.14), which means that
|j1→2|<|jc|<|j|=|j0→1|.(3.3.3)
Consequently, F1(t) will increase towards higher values due to (3.1.11) until it even-
tually reaches F1≈Fh(j)2. If on the other hand jis smaller than jc(magenta line
in Fig. 3.14) F1(t) will decrease for complementary reasons (magenta squares). It is
now apparent that the choice of the external current jin comparison to the inter-
section point jcis crucial. From the above discussion we come to the conclusions
that
|j|>|jc| ⇒ high field at emitter, i.e. F1(t0) ≈Fh(j),(3.3.4)
|j|<|jc| ⇒ low field at emitter, i.e. F1(t0) ≈Fl(j).(3.3.5)
We may now argue that (3.3.4) and (3.3.5) are still approximately valid, even if
the initial conditions (3.3.2) are not fulfilled. Let us consider a superlattice at a fixed
external current j, which initially contains an accumulation front at a position pa
far away from the boundary, and possibly further fronts at positions p > pa. Then
the region to the left of paincluding the emitter region is in the low field domain.
If jis larger than jcthe emitter region is required to be at a high field by (3.3.4).
This apparent “conflict” can be resolved by the dynamic generation of a new charge
depletion front at the emitter. A converse argument applies for the generation of
an accumulation front. The preliminary rules for the front generation can therefore
be summarized by
GI Generate accumulation front at emitter, if |j|<|jc|and if the leftmost front
is a depletion front.
GII Generate depletion front at emitter, if |j|>|jc|and if the leftmost front is an
accumulation front.
In Fig. 3.15 we checked numerically that the approximations leading to rules GI,
GII are justified, by examining the front generation for different values of jand
σ. We see that the conditions for depletion front generation can be accurately
predicted by GII. In the case of the generation of accumulation fronts, GI can only
be checked for currents |j|<|js
min|(Fig. 3.5), since otherwise the newly generated
front has zero or negative velocity and will not detach from the emitter.
32
3.3 Front Generation and Annihilation
0
1
2
el. density (nm -ND) [ND]
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
10 20 30 40
well index m
0
1
2
-6
-3
0
electric field [MV/m]
-6
-3
0
-6
-3
0
-6
-3
0
-6
-3
0
-6
-3
0
-6
-3
0
-6
-3
0
10 ps
40 ps
100 ps
700 ps
1300 ps
1900 ps
60 ps
300 ps
Figure 3.16: Electron density and electric field profiles for superlattice B with σ=
1 Ω−1m−1at different points in time for the first 40 wells. For 0 ≤
t≤40 ps the emitter is in the low field domain, i.e. the leftmost front
(not plotted) is an accumulation front and |j|>|jc|. A depletion front
starts to form at the emitter (blue squares). At t= 40 ps the external
current is switched to |j|<|jc|and the depletion front retracts to the
emitter and the front generation is not successful.
Rules GI, GII only apply if the leftmost front is already fully detached from
the emitter. Otherwise the newly generated front can annihilate a nearby front of
opposite polarity. This may occur in the common scenario of a dipole injection as
shown in Fig. 3.16. Here for t < 40 ps the conditions of rule GII are fulfilled, and a
depletion front starts to form. But before the depletion front is fully developed, we
switch the external current, such that GI applies. We see that in this case the half
formed depletion front retracts to the emitter contact. This is in contrast to the
scenario in Fig. 3.17, where the switching to the conditions of GI occurs at t= 60 ps.
By that time, the depletion front has reached a critical size, which allows it to be
detached from the emitter. Together with the subsequently generated accumulation
front, a dipole is generated.
The rule GI for the generation of an accumulation front should therefore be
modified, to require that the leftmost depletion front is at least ph≈2daway from
the emitter and a similar parameter plshould be introduced into GII. The revised
2F1is not exactly equal to Fh(j), since for t > 0 we have n1> ND. Therefore j1→2does not
obey the homogeneous characteristic and its high field intersection point with the external
current jwill be between Fh
min and Fh(j).
33
3 Front Dynamics in One Spatial Dimension
0
1
2
el. density (nm -ND) [ND]
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
10 20 30 40
well index m
0
1
2
-6
-3
0
electric field [MV/m]
-6
-3
0
-6
-3
0
-6
-3
0
-6
-3
0
-6
-3
0
-6
-3
0
-6
-3
0
10 ps
40 ps
100 ps
700 ps
1300 ps
1900 ps
60 ps
300 ps
Figure 3.17: Same scenario as in Fig. 3.16, but jis switched at t= 60 ps. A dipole,
consisting of a leading depletion front and a trailing accumulation front
is successfully injected at the emitter.
rules for front generation at the emitter then read:
GI0Generate accumulation front at emitter, if |j|<|jc|and of the leftmost front
is a depletion front which is at least at position ph.
GII0Generate depletion front at emitter, if |j|>|jc|and if the leftmost front is an
accumulation front which is at least at position pl.
We may now reexamine the scenario in Fig. 3.1(b). At t= 157 ns we have a
dipole configuration with a leading depletion and a trailing accumulation front.
The current is therefore j=jd. At t= 160 ns the depletion front reaches the
collector, i.e. Nd= 0. Then (3.2.2) requires that the velocity of the remaining
accumulation front drops to zero, and at the same time the current rises sharply
due to Fig. 3.13. Eventually we have |j|>|jc|= 2.6 A/mm2and a depletion front
is injected at the emitter by GII0. After that jstarts to drop towards jd, but as
soon as |j|<|jc|, and the depletion front has traveled by ph, the conditions for GI0
are fulfilled, and a new accumulation front is injected at the emitter at t= 161 ns.
All in all we see that the system responds to the event that the first depletion
front hits the collector, by the generation of a dipole with a leading depletion and
trailing accumulation front at the emitter. For the resulting tripole configuration,
the current j(2,1) is required (see Fig. 3.13). Since |j(2,1)|<|jc|and the leftmost
front is an accumulation front, no new fronts will be generated (see GI0,GII0), and a
34
3.3 Front Generation and Annihilation
0
1
2
el. density (nm -ND) [ND]
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
50 60 70 80
well index m
0
1
2
-6
-3
0
3
electric field [MV/m]
-6
-3
0
3
-6
-3
0
3
-6
-3
0
3
-6
-3
0
3
-6
-3
0
3
-6
-3
0
3
-6
-3
0
3
4200 ps
4300 ps
4500 ps
4700 ps
4800 ps
4900 ps
4400 ps
4600 ps
Figure 3.18: Electron density and electric field profiles for a collision and annihila-
tion process of a fast depletion front with a slow accumulation front.
Parameters: σ= 1 Ω−1m−1;j=−3 A/mm2.
current plateau with j=j(2,1) is maintained until the rightmost accumulation front
hits the collector at t= 163 ns. The current then drops to jd, but no new front is
generated at the emitter, until the cycle starts over again with the next depletion
front reaching the collector at t= 170 ns.
3.3.2 Front Collisions
From the current velocity characteristic Fig. 3.13 and from (3.2.2) we conclude that
the accumulation and depletion fronts may move at different velocities. This opens
up the possibility for a collision of two fronts with opposite polarity, and may lead
to interesting scenarios. Such a collision is shown in Fig. 3.18. We see that both
fronts annihilate each other, as can be expected from the fact that Qa=−Qd.
A basic example, where the collision of opposite fronts plays a role, is shown in
Fig. 3.19. This scenario resembles Fig. 3.1(b), since it also exhibits a tripole con-
figuration with two accumulation and one depletion fronts with the corresponding
current plateau j=j(2,1) (see Fig. 3.13). But now the fast depletion front catches up
with the rightmost accumulation front before it reaches the collector, and those two
fronts annihilate. The remaining accumulation front now triggers the generation of
a dipole at the emitter, as explained in the previous subsection for Fig. 3.1(b). At
any time we have an odd number of fronts present in the system, which explains,
why the current does not reach jd.
35
3 Front Dynamics in One Spatial Dimension
well
1
100
well
1
100
Figure 3.19: Front evolution as in Fig. 3.1, but for U= 6 V and σ= 1.3 Ω−1m−1.
3.3.3 Front Annihilation at the Collector
A further rather unspectacular elementary process is a front reaching the collector.
Such a front gets absorbed in the contact and vanishes from the system, thereby
reducing Naor Ndby one.
This concludes the set of elementary front processes, and we will see in the next
chapter, how they can be deployed to describe a large part of the dynamical bi-
furcation scenarios found in superlattices. We stress however that here we only
considered fully developed fronts and new interesting scenarios are expected, if we
also take into account partly developed fronts. Since such fronts are not individu-
ally stable, and often occur in combination with other fronts, an analysis of such
phenomena is rather complicated, and more likely to be specific for one particular
set of parameters.
36
4 Chaotic Front Dynamics
Simplification good!
Oversimplification bad!
(Larry Wall)
In Chapter 3 we have studied the basic building blocks for the front dynamics in
one spatial dimension. In this chapter we will examine how those elements can be
combined to yield interesting bifurcation scenarios, including chaos. While chaotic-
ity in periodically driven superlattices has been extensively studied theoretically
[74, 75, 76, 77, 78] and experimentally [79, 80] we will concentrate on the ques-
tion, how chaotic behavior can be obtained under fixed external voltage conditions
[81, 23].
For a fundamental understanding of the underlying bifurcations we will introduce
the front model, which retains the basic bifurcation structure, but is much easier to
handle numerically and analytically.
4.1 Bifurcation Scenarios of the Microscopic Model
We consider an N= 100 period superlattice of type B (cf. Table 2.2 on page 12),
and use the external voltage Uand the contact conductivity σas the bifurcation
parameters. From the discussion in Sec. 3.3.1 we learned that σgoverns the injection
of fronts at the emitter contact via the critical current jc(σ) (Fig. 3.15). Since the
resulting bifurcation scenarios are complicated, we will first study the particular
case of σ= 0.5 (Ωm)−1, and later consider the necessary modifications for general
σ.
4.1.1 The Case σ= 0.5 (Ωm)−1
For σ= 0.5 (Ωm)−1we have |j(1,2)|<|jc(σ)|<|jd|(c.f Fig. 3.13). If we vary
the applied voltage U, we typically observe front patterns as in Fig. 4.1,which are
reminiscent of the chaotic front dynamics in the Gunn-diode [82]. For a small voltage
(U= 0.50 V) we observe that the fronts are generated as dipoles at the emitter, with
a leading accumulation and a trailing depletion front. The leading accumulation
front catches up with the depletion front of the preceding dipole, and the two fronts
merge and annihilate at exactly the same position in each cycle. This scenario
corresponds to the one in Fig. 3.19, with the role of the accumulation and depletion
37
4 Chaotic Front Dynamics
1.00 V
1.20 V
1.80 V
2.00 V
150 200
Time [ns]
1.05 V
0.50 V
0.70 V
0.82 V
0.90 V
0.98 V
Figure 4.1: Dynamic evolution of the charge density for various voltages at σ=
0.5 Ω−1m−1in a superlattice of type B with N= 100 wells (space-time
plots). Regions of electron accumulation and depletion are denoted by
blue and red, respectively. In each panel the emitter (collector) is located
at the lower (upper) edge.
38
4.1 Bifurcation Scenarios of the Microscopic Model
fronts reversed. With increasing U, we observe what appears to be a period doubling
cascade, with two (U= 0.70 V in Fig. 4.1) and four (U= 0.82 V) alternating
positions where the front annihilation occurs. A further increase in the voltage yields
irregular behavior (U= 0.90 V, 1.00 V, 1.20 V) interrupted by periodic windows
(U= 1.05 V). For even higher voltages, the fronts may occasionally reach the
collector, but even then the interchange between chaotic (U= 1.80 V) and periodic
(U= 2.00 V) regimes persists.
The chaotic behavior can also be observed in the experimentally accessible current
trace, as demonstrated in Fig. 4.2. Here we observe a further interesting feature,
namely that not all fronts are fully developed. One example can be seen at t= 155 ns
in the electron density plot of Fig. 4.2 (see also the case U= 1.2Vin Fig. 4.1). Here
an accumulation front seems to detach from the emitter, but instead of catching
up with the leading depletion front, it merges with a new depletion front from
the emitter, before either of the fronts can be considered as fully developed. Such
compositions of partly developed fronts can not be described in the framework
of single stable fronts, which was developed in Chapter 3. In particular they do
not obey the current velocity characteristic of Fig. 3.13. Such composite front
phenomena, resemble the excitons in solid state physics, since they often appear
in pairs without net charge, and form a bound state with limited life time. Their
dynamics may be treated by a yet to be developed correlated front theory, which
is however beyond the scope of the present work. In the following we will refer to
this kind of phenomena as excitonic fronts. Similar effects also appear for pulses in
excitable media [83].
The difference between periodic and chaotic behavior is also illustrated by the
phase portraits as shown in Fig. 4.3, which show the system trajectory in the phase
space projected onto the subspace defined by n10 and n20. In Fig. 4.3(a) the tra-
jectory in the phase space is complicated, but still periodic, while in the chaotic
regime in Fig. 4.3(b) the trajectory is aperiodic.
The full bifurcation scenario is shown in Fig. 4.4(a), where for each voltage U
the set of front annihilation positions {pc}is plotted. Here we may interpret a
discrete set of pc’s for a given voltage as an indication for periodic behavior (for
instance the four points at U= 0.82 V, which correspond to a period four orbit),
while a continuous set of collision points is an indication of chaotic behavior (cf.
U= 0.90 V).
Starting from low voltages, we observe a period doubling bifurcation with periods
1, 2 and 4 in the regions A, B, and C of Fig. 4.4(a), respectively. The following region
D contains two chaotic bands at its boundaries, which are separated by a period
six orbit. While the chaotic band at the left edge of region D is rather narrow, the
band at the right edge is comparatively broad. The most striking feature in region
D is the center of a crossing of at least three straight lines, which in the following
will be called a cobweb structure. In this case the cobweb is located in the chaotic
band at the right edge of region D. This chaotic band ends with the transition to
the period 4 behavior in region E. The following region F is again chaotic, and
39
4 Chaotic Front Dynamics
well
1
100
well
1
100
Figure 4.2: Electron density (upper panel), electric field (middle panel) and cur-
rent evolution (bottom panel) in the chaotic regime. Parameters as in
Fig. 4.1, but with U= 1.15 V. The color code is explained in Fig. 3.1.
The black current trace in the bottom panel is the running average of
the raw current data (cyan line) over an interval of 0.5 ns.
-1 0 1 2
(n10- ND) [ND]
-1
0
1
2
(n20- ND) [ND]
-1 0 1 2
(n10- ND) [ND]
-1
0
1
2
(n20- ND) [ND]
a) b)
Figure 4.3: Phase portrait of the electron densities n20 vs. n10 for superlattice pa-
rameters as in Fig. 4.1 for time series from t= 50 ...200 ns. (a) periodic
behavior at U= 0.82 V, (b) chaotic behavior at U= 1.15 V.
40
4.1 Bifurcation Scenarios of the Microscopic Model
is bounded by the larger period three region G. In regions A to G we observe a
number of continuous and almost straight lines, which exist across various regions,
even in the chaotic regimes. These lines also give rise to the cobweb structure with
its center in region D.
In the following voltage interval H in Fig. 4.4, we observe collisions close to the
emitter. They are the footprints of the annihilation of excitonic fronts, as discussed
before. Note however that the numerical method for collision detection only works
reliably for well numbers m > 5, which may limit our ability to detect excitonic
collisions which occur very close to the emitter. In region I fronts occasionally reach
the collector, and we observe a dynamics with seven distinct collision points. The
excitonic collisions are suppressed in this region, but they reappear in region J,
where we have fronts reaching the collector and excitonic collisions.
In principle, the position of collision pcis a real number, but in practice it is
difficult to determine pcwith an error which is less than the width of the accumula-
tion front (cf. Fig. 3.18). To distinguish between chaotic and periodic behavior, we
may therefore consider a suitable Poincar´e section of one of the continuous dynam-
ical system variables. This is shown in Fig. 4.4(b) for the time difference between
two consecutive maxima of the electron density in well 20, n20(t). This bifurcation
shows the chaotic bands at the same locations as in Fig. 4.4(a) (note however the
different voltage scale). We observe that chaotic and periodic behavior alternate up
to a voltage of about U= 3.6 V, which corresponds to the case where about half
of the superlattice is in the high field regime. For U > 3.6 V, the chaoticity sud-
denly disappears. From Fig. 4.5 we see that the reason for this change is associated
with the transition from an operation mode, in which every third high field tongue
reaches the collector (U= 3.55 V in Fig. 4.5) to a mode where only every second
high field tongue reaches the collector (U= 3.65 V). For even higher voltages, no
collisions occur, and all fronts reach the collector (cf. U= 5.0 V).
4.1.2 Varying σ
Now that we have an idea of the bifurcations appearing for σ= 0.5 (Ωm)−1, we
proceed to the case of general contact conductivity. As we have learned in Sec. 3.3.1,
the parameter σgoverns the injection of fronts at the emitter. From the analysis
of the case σ= 0.5 (Ωm)−1we see that the excitonic regimes (regions H and J in
Fig. 4.4(a) complicate the analysis, and it would be nice if we could avoid them.
This is addressed by choosing a slightly lower contact conductivity σ= 0.45 (Ωm)−1,
which leads to a lower critical current density, such that the condition |j(1,2)|<
|jc(σ)|<|j(2,3)|holds. Furthermore, since the simplified models we will propose
below are most successful in the regimes where no fronts reach the collector, we
may also choose a longer superlattice. In Fig. 4.6 the bifurcation diagram for a
σ= 0.45 (Ωm)−1and N= 200 well superlattice is shown. We see that the regions
A to G show the same behavior as the corresponding regions in Fig. 4.4(a). However
the excitonic regions have disappeared, and instead we observe a period 6 regime
41
4 Chaotic Front Dynamics
a)
b)
0.5 11.5 2
U [V]
20
60
100
Well #
A B
C
D
E
F G H I J
0.5 11.5 2
Voltage [V]
0
2
4
∆t [ns]
1 2 345 6
Voltage [V]
0
2
4
6
∆t [ns]
Figure 4.4: (a) Positions where accumulation and depletion fronts annihilate vs volt-
age at σ= 0.5 Ω−1m−1. The color scale indicates high (blue) and
low (white) numbers of annihilations at a given well. (b) Time differ-
ences between consecutive maxima of the electron density in well no.
20 (n20(t)) vs voltage at σ= 0.5 Ω−1m−1. Time series of length 600 ns
have been used for each value of the voltage. The inset in (b) shows the
time differences for a larger voltage range. Source: [81, 23].
42
4.1 Bifurcation Scenarios of the Microscopic Model
5.0 V
150 200
Time [ns]
4.5 V
2.4 V
3.55 V
3.65 V
Figure 4.5: Dynamic evolution of the charge densities (upper panels) and electric
fields (lower panels) for various voltages. Parameters are as in Fig. 4.1,
color code as in Fig. 3.1.
43
4 Chaotic Front Dynamics
1 2 3 4 5
U[V]
0
50
100
150
200
Well #
A B C
E
D F G I
H
J L
K
Figure 4.6: Bifurcation diagram as in Fig. 4.4(a), but with N= 200 and σ=
0.45 Ω−1m−1. From [84]. Other parameters as superlattice B (Ta-
ble 2.2).
in region H. Region I shows chaotic behavior, except for a small period 7 band at
its center. Furthermore we find a second cobweb structure in region I, which shares
its horizontal line with the first cobweb in region D. The next region J has period
5. This is followed by a small chaotic region K, before the fronts start to reach the
collector in region L. Note how again straight continuous lines run through the whole
bifurcation diagram, and are then inflected as they reach the K region. The origins
of the rich bifurcation scenario apparent in Fig. 4.6, including the chaotic bands,
the cobweb structures and the sequence of the various periods will be explained by
analytical considerations in Chapter 5.
We have seen that a small variation in σhas already a nontrivial effect on the
bifurcation diagrams (Fig. 4.4 vs. Fig. 4.6). We may now ask, how the bifurcation
diagram changes, as we further vary the contact conductivity. Since the calculation
of the bifurcation diagrams is time consuming, we again use the short superlattice,
with N= 100 wells. An overview of the different bifurcation scenarios for varying
σis given in Fig. 4.7. We note that for σ= 0.4 (Ωm)−1the scenario resembles the
situation in regions A and B of Fig. 4.4, which corresponds to the periodic tripole
configurations as in the first two panels of Fig. 4.1. We find the well known cob-
web structure for σ= 0.45,...,0.52 (Ωm)−1, which however shifts to lower voltages
and lower well numbers as σincreases. For σ= 0.54 (Ωm)−1, the scenario seems
to have fundamentally changed. The cobweb has disappeared the collisions close
44
4.1 Bifurcation Scenarios of the Microscopic Model
to the emitter indicate the presence of excitonic fronts. Also the familiar period
three window has disappeared, and instead we find a large period four window, with
three collision points in the sample, and every fourth front reaching the collector.
A small increase to σ= 0.55 (Ωm)−1again changes the bifurcation diagram com-
pletely. Fronts reach the collector already at voltages below 1V, and at the same
time front collisions take place close to the emitter. There are now only very few
continuous lines present, and the whole structure appears to be washed out. This
trend continues for σ= 0.57 (Ωm)−1. The bifurcation diagram for σ= 0.60 (Ωm)−1
is missing in Fig. 4.7. The reason is that in this case no collisions within the sample
occur. We have jc≈jd, which means that we are at the symmetry point, where
accumulation and depletion fronts have equal rights. At any time there are two
fronts in the sample, which move in parallel, until the leading front reaches the
collector and reappears at the emitter.
By further increasing σ, we enter the regime, where the depletion fronts are faster
than the accumulation fronts. In this case it is numerically more difficult to de-
tect the position of the annihilation with high accuracy. Thus the lines in Fig. 4.7
for σ≥0.65 (Ωm)−1are in general broader than before. Nevertheless the cob-
web structure at σ= 0.8 (Ωm)−1is clearly visible, and also somewhat weaker for
σ= 0.75 (Ωm)−1. These cobwebs resemble the cobweb found at σ= 0.45 (Ωm)−1
(Fig.4.6), but is flipped along the voltage axis. This is a consequence of the sym-
metry transformation, which we discuss below. Another interesting feature is the
reconnection of the period doubling bifurcation, which occurs at σ= 0.7 (Ωm)−1
and σ= 0.65 (Ωm)−1, and causes a distinct bubble like structure in the bifurcation
diagram.
It is now interesting to plot a “phase-diagram” of chaotic behavior in the (U, σ)-
plane as shown in Fig. 4.8, which was obtained by considering the autocorrelation
function C(τ) = hn20(t)n20(t+τ)it[23]. For periodic behavior C(τ) does not decay
even for large values of τ > 20 ns, while for chaotic behavior C(τ) decays with a
correlation time less than 20 ns. We note that chaotic behavior is only possible,
if we choose σsuch that |j(1,2)|<|jc(σ)|<|j(2,1)|. Furthermore there exist two
larger disjoint regions which are very roughly “point symmetric” about a point at
(U≈3.5 V, σ ≈0.6 Ω−1m−1). The origin of this “symmetry” is the approximate
invariance of the system under the simultaneous permutation of accumulation with
depletion fronts, and low field with high field domains.
4.1.3 Lyapunov Exponents
To further confirm the chaoticity, the largest Lyapunov exponent λfor U≈1.15V,
σ≈0.5 Ω−1m−1) (Fig. 4.2) was calculated for a long (t= 0 ...100 µs) time series
of n20(t) [23] using the Wolf algorithm [85]. The result λ= 1.1×109s−1is a clear
indication of chaos.
45
4 Chaotic Front Dynamics
4.5 5 5.5 6 6.5
U[V]
0
20
40
60
80
100
Well #
σ=0.9 Ω−1m-1
4.5 5 5.5 6 6.5
U[V]
0
20
40
60
80
100
Well #
σ=0.70 Ω−1m-1
0.5 1 1.5 2
U[V]
0
20
40
60
80
100
Well #
σ=0.55 Ω−1m-1
0.5 1 1.5 2
U[V]
0
20
40
60
80
100
Well #
σ=0.51 Ω−1m-1
4.5 5 5.5 6 6.5
U[V]
0
20
40
60
80
100
Well #
σ=0.8 Ω−1m-1
4.5 5 5.5 6 6.5
U[V]
0
20
40
60
80
100
Well #
σ=0.65 Ω−1m-1
0.5 1 1.5 2
U[V]
0
20
40
60
80
100
Well #
σ=0.54 Ω−1m-1
0.5 1 1.5 2
U[V]
0
20
40
60
80
100
Well #
σ=0.45 Ω−1m-1
4.5 5 5.5 6 6.5
U[V]
0
20
40
60
80
100
Well #
σ=0.75 Ω−1m-1
0.5 1 1.5 2
U[V]
0
20
40
60
80
100
Well #
σ=0.57 Ω−1m-1
0.5 1 1.5 2
U[V]
0
20
40
60
80
100
Well #
σ=0.52 Ω−1m-1
Well #
0.5 1 1.5 2
U[V]
0
20
40
60
80
100 σ=0.4 Ω−1m-1
Figure 4.7: Bifurcation diagrams as in Fig. 4.4(a), for various σ.
46
4.2 The Front model
1 2 3 4 567
U [V]
0.4
0.6
0.8
1
σ [(Ωm)-1]
Figure 4.8: Two parameter bifurcation diagram. Black squares: chaotic behavior;
green shading: periodic oscillations; white region: absence of oscilla-
tions. Source: [23].
4.2 The Front model
We are now in a position to approximate the microscopic dynamics of the elec-
tron densities niby means of a simple front model, in which the positions of the
accumulation fronts a1. . . aNaand the depletion fronts d1. . . dNdand the overall
current jare the new dynamical variables. Here Naand Nddenote the number of
accumulation and depletion fronts in the system. We will see that this step from
the microscopic description to a front description does not only greatly reduce the
dimensionality of the system, but also the number of physical parameters. The in-
troduction of front positions was already shown to be useful in Sect. 3.2 for the case
of the free motion of noninteracting fronts far away from the boundaries. We now
make the assumption that the essential dynamics of the system can be described
in terms of front positions, even if the fronts are close to each other or close to the
boundaries. Such a “dilute gas” approximation will obviously fail if the density of
fronts is large or if the typical time scale for interactions between fronts can not be
assumed to be small.
4.2.1 Elimination of the current density
Let us define by
Lh(j) = −U−LF l(j)
Fh(j)−Fl(j)≈−U
Fh(j)(4.2.1)
the partial length of the superlattice which is in the high field region. Here L=
Nd is the total length of the superlattice, and in the last step we have used the
47
4 Chaotic Front Dynamics
approximation Fl≈0. From (3.1.7) and (3.0.3) it follows that Lhimposes a global
constraint on the front positions by
Lh(j) =
Nd
X
i=1
di−
Na
X
i=1
aimod L. (4.2.2)
The expression mod Lin (4.2.2) means that Lhas to be added if aNa> dNasuch
that Lh∈[0, L]. We stress that due to our center-of-mass-like definition of the
front positions, (4.2.2) is exact, even for fronts with a finite width. In particular,
the discreteness of the superlattice does not play a role here.
Differentiating (4.2.1) and (4.2.2) with respect to tfor U= const yields
∂Lh
∂t =−U
(Fh)2
∂F h
∂j
∂j
∂t ,(4.2.3)
and (using (3.1.8))
∂Lh
∂t =Ndvd−Nava,(4.2.4)
respectively. By combining (4.2.3) and (4.2.4) we obtain the evolution equation for
the current density,
∂j
∂t = (Ndvd(j)−Nava(j)) Fh(j)2
U∂F h
∂j
.(4.2.5)
From (4.2.5) it follows that the current will relax to a state, where (3.2.2) is fulfilled,
i.e. vd(j)
va(j)=Na
Nd
.(4.2.6)
From the denominator in (4.2.5) we note that this relaxation will be fast, if Uand
∂F h
∂j are small. For example we may consider the relaxation towards the dipole
domain current density jdwith Nd=Na= 1. In linear approximation we have
va/d(j)≈va/d(jd) + (j−jd)∂jva/d(jd).(4.2.7)
From (3.1.14) we obtain ∂jvd=d/(eND). From Fig. 3.13 we may further approxi-
mate (for this particular superlattice only) ∂jvd(ja)≈ −∂jvd. Using va(jd) = vd(jd)
we get for the first factor in (4.2.5),
(Ndvd(j)−Nava(j)) ≈(j−jd) (∂jvd(jd)−∂jva(jd)) ≈(j−jd)2d
eND
.(4.2.8)
Since this factor vanishes at j=jd, the leading contribution from the second factor
in (4.2.5) is in zeroth order of (j−jd). Using (4.2.1) we arrive at
1
jd
∂j
∂t ≈ −j−jd
jd
2d
eND
Fh(jd)
Lh∂jFh(jd)≈ −j−jd
jd
d
Lh
1
τeff
,(4.2.9)
48
4.2 The Front model
with τeff ≈1 ps. Since Lh/d < N, which is the number of wells in the high field
domain, we obtain typical relaxation times of less than 100 ps. During this time,
the fronts typically travel less than two wells, which justifies the simplification that
the current relaxation according to (4.2.9) is almost instantaneous. This means
that (4.2.6) is always immediately fulfilled. By formally inverting the left hand side
of (4.2.6) and taking into account the results of Sec 3.2 we arrive at the conclusion
that
j=j(Na,Nd)=jNa
Nd,(4.2.10)
which is a monotonically increasing function, since vd(va) is monotonically increas-
ing (decreasing). We can therefore replace the condition |j|<|jc|appearing in rule
GI0on page 34 by an equivalent condition
Na
Nd
< rc,(4.2.11)
where the parameter rcis defined by jc=j(rc). A similar statement applies to rule
GII0. We have therefore managed to enslave the current density jto the fraction
Na/Nd. Note that in particular jd=j(1).
4.2.2 The rules for the front model
For the analysis of the bifurcation scenario, it is sufficient to consider the dynamics
in the Poincar´e section which is defined by the hyperplane, where Naor Ndchange.
The absolute time between such events is not important, and we are therefore free
to rescale the velocities to our convenience. In the following we rescale time such
that va+vd= 2, which together with (4.2.6) gives the front velocities as
va=2Nd
Na+Nd
, vd=2Na
Na+Nd
.(4.2.12)
We require that the fronts evolve according to (4.2.12), until an event which
changes the number of fronts occurs. Such an event may be the generation of a new
front at the emitter according to the rules GI0and GII0as described in Sec. 3.3.1.
Furthermore two fronts can collide as described in Sec. 3.3.2, which will simply
eliminate the corresponding diand aifrom the system of variables and decrease Na
and Ndaccordingly by one. The third possibility is the annihilation of a front at
the collector as described in (Sec. 3.3.3). We may summarize the complete front
model by the following set of rules:
FI The positions of the accumulation fronts aifor i= 1 . . . Naand depletion
fronts difor i= 1 . . . Ndevolve according to ˙ai=vaand ˙
di=vdwith the
velocities (4.2.12) until one of the following rules applies.
49
4 Chaotic Front Dynamics
FII If Na/Nd< rcand ph< d1< a1then increase Naby one, re-index ai→ai+1
for all iand set a1= 0 (injection of accumulation front).
FIII If Na/Nd> rcand pl< a1< d1then increase Ndby one, re-index di→di+1
for all iand set d1= 0 (injection of depletion front).
FIV If ai0=dj0for any i0, j0then decrease Naand Ndby one, re-index ai+1 →ai
for i≥i0and dj+1 →djfor j≥j0(annihilation of front pair).
FV If aNa> L decrease Naby one (accumulation front hits collector).
FVI If dNd> L decrease Ndby one (depletion front hits collector).
Here phand plare the phenomenological distance parameters from GI0and GII0on
page 34, which suppress the front generation for d1≤phand a1≤pl, respectively
[86].
The only parameters appearing in the front model are rc,phand pl, which govern
the generation of new fronts at the emitter and L, which influences the annihilation
at the collector. The voltage parameter Lhis connected to the voltage by (4.2.1),
and in principle also depends weakly on Na/Nddue to (4.2.10), but for simplicity
we consider Lhto be constant. Then Lhonly enters in the initial condition for the
front positions (see Eq. (4.2.2)). These five parameters should be contrasted to the
large set of parameters of the microscopic model (Tables 2.1 and 2.2). However, in
particular phand plmight be difficult to derive quantitatively from the microscopic
model, and should rather be regarded as fit parameters. Again phand plcan in
principle depend on Naand Nd, but for simplicity we assume them to be constant.
The dynamical variables of the systems are the positions diand aiof the fronts.
Due to the constraint given by (4.2.2), the number of degrees of freedom is then given
by fd=Na+Nd−1 and we have fd≤Nmax
a+Nmax
d−1 = 2n−2, which in general
is much smaller than N, the number of degrees of freedom of the full microscopic
model. It is however a peculiarity of this system that fdchanges dynamically. To
avoid the mathematical complications that arise from the fact that the number of
dynamical system variables is not constant, we formally extend the arrays of front
positions to the maximal possible size, a1,...aNmax
aand d1,...dNmax
dand consider
additionally Naand Ndas discrete system variables. The new additional front
positions aNa+1 . . . aNmax
aand dNd+1 . . . dNmax
ddo not appear in the front rules FI to
FVI on page 49 and we can just set them to zero for definiteness. By this formal
transformation we have now obtained a system with Nmax
a+Nmax
dcontinuous and
two discrete variables1. This type of system therefore belongs to the mathematical
class of hybrid systems. Hybrid models are of fundamental interest in the field of
theoretical computer science, where they are used to describe the interaction of a
digital (i.e. discrete) computer with an analog environment [87].
1Since the product of two countable sets is also countable, we may as well replace the two discrete
variables by only one.
50
4.2 The Front model
rcn Nmax
aNmax
d
0 1 0 1
(0,1
2]2 1 2
(1
2,2
3]3 2 3
(2
3,3
4] 4 3 4
(n−2
n−1,n−1
n]n n-1 n
1∞ ∞ ∞
[n
n−1,n−1
n−2) n n n-1
[3
2,2) 3 3 2
[2,∞) 2 2 1
∞1 1 0
Table 4.1: Maximum possible number of accumulation and depletion fronts and the
number of necessary tanks n(see Sec. 5.2) for various values of rc.
Note that the rules of the front model are invariant under the simultaneous trans-
formation of
ai↔di, Na↔Nd, ph↔pl,(4.2.13)
rc→1
rc
, Lh→L−Lh,
i.e. accumulation and depletion fronts are exchanged, rcis inverted, and the high
field and low field domains are exchanged ( Lh→L−Lh). This exact symmetry
can therefore explain the qualitative point symmetry found in Fig. 4.8, since the
transition rc→r−1
cinduces a corresponding transformation σ(j(rc)) →fs(σ(j(rc)))
with the fixed point σ(jd) = fs(σ(jd)).
In view of the symmetry of (4.2.13) we may restrict our analysis to the case
rc<1. We furthermore set pl= 0 since the accumulation fronts are rather narrow
and should not suppress the generation of a trailing depletion front. In this case
rule FIII always applies, if the first front is an accumulation front, since in this case
Na/Nd≥1 and injects a new depletion front. On the other hand rule FII can only
apply if Nd=Na+ 1. It does not apply as long as Na> rc/(1 −rc) and Nacan
then only decrease, since FII is the only process which generates new accumulation
fronts. Consequently rcimposes the following limits on the number of fronts:
Nd≤n, Na≤n−1,(4.2.14)
where nis the largest integer less than 1/(1 −rc) + 1. The dependence of the
maximum front numbers Nmax
aand Nmax
don rcis summarized in Table 4.1.
Once the conditions for FII are fulfilled and an accumulation front is injected,
it is immediately followed by the injection of a depletion front due to rule FIII.
51
4 Chaotic Front Dynamics
Effectively we therefore inject a pair of fronts, i.e. a dipole, with a leading accumu-
lation and a trailing depletion front. In the language of field domains, this process
detaches a high field domain from the emitter and opens a new one.
In Table 4.1 we also list the parameter n, which is defined as
n= max [Nmax
a, Nmax
d].(4.2.15)
Since nis invariant under the symmetry transformation (4.2.13), we propose that n
will be a suitable parameter for classifying different bifurcation behaviors. Indeed
we will see in Chapter 5 that ncorresponds to the number of tanks, which are
necessary to describe a given dynamics.
4.2.3 The case n= 3
The numerical integration of the front model is facilitated by the fact that the evo-
lution of the front positions is piecewise linear due to FI with the velocities given in
(4.2.12). We can therefore calculate the times tFII,...,tFVI, until the corresponding
conditions in FII. . . FIV would be fulfilled under the assumption that Naand Nd
would not change. The actual event is then determined by the minimum time tFX,
with X = II ...VI. The fronts are then moved to the new positions a0
i=ai+tFXva
and d0
i=di+tFXvd, and the changes in the discrete variables Naand Ndare
performed as prescribed by the respective rule FX.
The numerical solution of the front model for pl= 0, and Nmax
a= 2 yields a
typical front pattern as in Fig. 4.9. We see that for small Lhthe front which is
closest to the collector, is always a depletion front. Since the fronts are generated
in pairs at the emitter, we have Nd=Na+ 1, and therefore va> vd. That means
that the accumulation fronts can catch up and annihilate with the respectively
preceding depletion fronts. This is the same behavior as observed in Sec. 4.1 for the
microscopic model. In fact the front pattern at low Lhin Fig. 4.9 can be directly
related to the ones in Fig. 4.1. As a particular striking example compare the period
seven orbits at Lh= 0.202 in Fig. 4.9 and at U= 0.98V in Fig. 4.1. As long as the
fronts do not reach the collector, the only relevant length scale for Lhis the distance
parameter ph. In the microscopic model, this parameter corresponds to the minimal
distance between the first depletion front and the newly generated accumulation
front and will in general depend on the buildup time of the accumulation front and
other microscopic parameters in a complicated way.
In its present form the front model is not chaotic, which is in contrast to the full
microscopic model. Instead arbitrarily long stable periodic orbits are possible. We
will discuss in the following Chapter 5, how chaoticity can be introduced in a generic
way. At higher values of Lh, we find the characteristic “tongues” (Lh= 0.595 in
Fig. 4.9), which also occur in the microscopic model (see U= 3.55 V in Fig. 4.5),
but we did not succeed in finding the other patterns in Fig. 4.5. One reason, why
the front model does not describe well the high Lhcase becomes apparent, if we
52
4.2 The Front model
Lh=0.090
Lh=0.140
Lh=0.160
Lh=0.181
Lh=0.202
Lh=0.209
Lh=0.289
Lh=0.595
Lh=0.971
time [arb. units]
70 75
Figure 4.9: Front evolution in front model for n= 3, pl= 0, ph= 0.115, rc= 0.51,
L= 1 and various values of Lh. Accumulation (depletion) fronts are
denoted by blue (red) lines. Lhcorresponds to the voltage Uin the
microscopic model (cf. Figs. 4.1, 4.5 ).
53
4 Chaotic Front Dynamics
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0
0.2
0.4
0.6
0.8
1
Position
0.16 0.18 0.2 0.22
0
0.2
0.4
0.6
0.8
1
Position
Lh
Figure 4.10: Bifurcation diagrams for positions of front collision vs Lhobtained
from the front model for n= 3 on two different scales. Parameters:
ph= 0.06, pl= 0, rc= 0.52, L= 1.0.
compare Lh= 0.971 in Fig. 4.9 and U= 4.5 V in Fig. 4.5, where fronts of opposite
polarity traverse the whole superlattice at only a very small distance to each other.
This is obviously not possible in the microscopic approach, since the fronts would
tend to annihilate. Thus the front model can still be improved in the high voltage
regime.
A further touchstone for the usefulness of the front model is given by its bifurca-
tion diagram as shown in Fig. 4.10. The prominent feature is again the cobweb-like
pattern at low voltages, which has a striking similarity with the corresponding pat-
terns in Fig. 4.4(a) and Fig. 4.6. In fact all regions from A to K of Fig. 4.6 can
also be identified in Fig. 4.9, only region K does not fit perfectly. In particular,
the vertical bands in Fig. 4.10(b) can be identified with the three chaotic bands
in the regions D and F of Fig. 4.6. However, since the front model, is not really
chaotic in this regime, they actually consist of ever finer subbands as shown in the
lower panel of Fig. 4.10. Apparently a period of more than seven different collision
points, appears chaotic in the microscopic model. Another feature of the original
54
4.2 The Front model
bifurcation scenario that is well reproduced by the front model is that the chaotic
behavior suddenly becomes periodic at about Lh= 0.53. On the other hand, we
do not observe periodic windows for Lh∈[0.36,0.53] which were present in the
microscopic model.
The fact that the topology of the nontrivial pattern up to the large period three
window U= 1.1V in Fig. 4.4 can be reproduced by the simple rules of the front
model, is a hint that such a pattern might be even more generic, as we will see in
Chapter 5.
We could now proceed to extract the detailed features of the bifurcation diagram
by a thorough analysis of the algebraic properties of the front model. For example
the horizontal lower line appearing in Fig. 4.10 is caused by ph. If Na= 1 and
Nd= 2, we can inject a new accumulation front by FII as soon as d1has reached
ph. As argued before, this will entail as well the injection of a depletion front, and
we have the situation:
Na= 2, Nd= 3, a1=d1= 0 d2=phd3−a2=Lh−ph.(4.2.16)
From (4.2.12) we get va−vd= 2/5. If now additionally Lh>2ph, it follows that
d3−a2> d2−a1and therefore the fronts d2and a1will be the first to annihilate.
If Lh>3/2ph, then the fronts d3and a2will be the first to annihilate, but by that
time d1> Phand therefore a new dipole is immediately injected at the emitter.
This maintains the velocities of the original d2and a1and in both cases the collision
occurs at a position
pz1 =ph
va
va−vd
=nph.(4.2.17)
For n= 3 and ph= 0.06 this yields the horizontal line in Fig. 4.10 at pz1 = 0.18.
A further analysis of the structure of the bifurcation diagram along these lines is
possible, but cumbersome. We will therefore in the next chapter introduce a model
which is better suited to an analytical approach.
4.2.4 Arbitrary n
In Fig. 4.11 the bifurcation diagrams of the front model for n= 4 and n= 5
are plotted. After the successful identification of many common features in the
bifurcation diagrams of the microscopic model and the front model for n= 3, we
would hope that at least some features from Fig. 4.11 also appear in one of the
panels of Fig. 4.7. However, this is apparently not the case. The reason for this
failure seems to be that with a large number of fronts, the approximation that
fronts can be considered as independent point-like “quasi-particles” breaks down.
In the language of statistical physics, the dilute gas approximation is no longer valid,
and we have to take into account three front interactions, and other complications.
We may speculate however that for very large superlattices with narrow fronts, a
bifurcation scenario as in Fig. 4.11 should arise.
55
4 Chaotic Front Dynamics
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0
0.2
0.4
0.6
0.8
1
Position
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0
0.2
0.4
0.6
0.8
1
Position
Lh
b)
a)
Figure 4.11: Bifurcation diagrams for positions of front collision vs Lhobtained from
the front model for (a) n= 4 (rc= 0.67) and (b) n= 5 (rc= 0.76).
Parameters: ph= 0.06, pl= 0, L= 1.0.
It is nevertheless still interesting to scan the (Lh, rc) plane of the front model
for regions of long periods, since they correspond to chaotic regimes of the micro-
scopic model. By varying rcand Lhsimultaneously we obtain the two parameter
bifurcation diagram of Fig. 4.12. Note that the broad horizontal bands in Fig. 4.12
are due to the fact that the changes in rcwithin the intervals given by Table 4.1
do not affect the dynamics of the system. The basic structure of the bifurcation
diagram obeys the symmetry of (4.2.13) and conforms well with the corresponding
bifurcation diagram of the microscopic model in Fig. 4.8.
56
4.2 The Front model
0 0.2 0.4 0.6 0.8 1
Lh
0.5
1
1.5
2
rc
Figure 4.12: Two parameter bifurcation diagram for the front model. Dark region
corresponds to (Lh, rc) pairs with at least 10 different points of front
collisions. Parameters: L= 1.0; for rc<1: ph= 0.06, pl= 0; for
rc>1: ph= 0, pl= 0.06. In the microscopic superlattice model, Lh
and rccorrespond to Uand σ, respectively (cf. Fig.4.8).
57
4 Chaotic Front Dynamics
58
5 The Tank Model
In the previous Chapter 4 we have introduced a simple front model, which aston-
ishingly well reproduces many features of the complex microscopic model, at least
in the low and intermediate voltage regimes, when no fronts reach the collector.
We will now further simplify the front model in this regime and will finally arrive
at a tank model. Such models have been extensively studied in computer science
and applied mathematics, since they describe the dynamic of production processes
[88, 89]. Typically one obtains a “strange billiard” behavior [90, 91], which means
that the system evolves piecewise linearly, and only changes its direction at the
boundary of a specific domain. The advantage of such an approach, is that these
type of models can often be treated analytically. As we will see, this simplification
allows us to relate the bifurcation scenario of the front system to the bifurcations
obtained in a simple low dimensional iterated map system. In the most simple
nontrivial case this map will be only one dimensional.
A connection between maps and single fronts has previously been studied in the
case of coupled map lattices [92] and for periodically driven systems [93]. In contrast
to those works, however, we are here concerned with the use of maps for a system
with interacting fronts [86].
5.1 Deduction from the Front Model
Let us now derive the tank model from the front model on page 49. The idea is
that instead of dealing with the position of accumulation and depletion fronts, we
restrict ourselves to the dynamics of the high field domains, which appear between
two fronts, or between the emitter and the first depletion front. Technically it is
again easier to start with the case pl= 0, but we will see that in principle, the tank
model even holds for general pl.
5.1.1 The Case pl= 0
We again assume rc<1 and for the moment pl= 0. Our first task is to derive a
condition, for which no fronts will reach the collector. We consider a situation where
Nd=Na+ 1 ≤n, at the point in time where a dipole is injected at the emitter by
the rules FII and FIII of the front model (see page 49). We then have a1= 0, d1= 0
and d2≤Lhby (4.2.2). From (4.2.12) we see that va−vd>2/(2n−1) and therefore
the time until a1and d2collide will be tcollision < Lh(n−1/2). On the other hand we
59
5 The Tank Model
d1
a2
a1
d2
d3
x3
x2
x1time
position
Figure 5.1: High field domain variables xiderived from front positions ai(blue lines)
and di(red lines). The orange shaded area denotes the high field domain.
have va<4/3 and therefore the time until a1reaches the collector is ttransit
a>3L/4.
We may then conclude that no fronts reach the collector if tcollision < ttransit
a, or
equivalently
Lhn−1
2<3
4L. (5.1.1)
For n= 3 and pl= 0, (5.1.1) states that for Lh<0.3Lno fronts will reach the
collector, which is confirmed by our simulation of the front model (Fig. 4.10). For
the rest of this section we assume that (5.1.1) is fulfilled.
The essential step in the derivation of the tank model, is that we now choose the
lengths of the high field domains,
x1=d1,(5.1.2)
xi=di−ai−1for i= 2 . . . Nd,(5.1.3)
as the new dynamical variables of the system (cf. Fig. 5.1). Here x1is special, since
it is the high field domain, which is connected to the emitter, and is therefore only
bounded by a depletion front. This is in contrast to all other high field domains
which are bounded by a depletion front from above and an accumulation front from
below. The introduction of the new variable xireduces the number of continuous
system variables from 2n−1 to n. We hereby lose the information on the position
of the high field domain within the superlattice. But the absolute front positions
do only occur in the rules FV and FVI of the front model, and they will not apply,
since we assumed that no domains will reach the collector. If pl= 0, the condition
Na=Nd−1 is always fulfilled and we need to keep track of only one discrete
60
5.1 Deduction from the Front Model
variable Nd. The global constraint (4.2.2) is translated to the new variables by
Lh=
Nd
X
i=1
xi.(5.1.4)
From the front velocities (4.2.12) we may obtain the shrinking and growing velocities
of the high field domains by
˙xi=(vd=2Nd−2
2Nd−1if i= 1,
vd−va=−2
2Nd−1else.(5.1.5)
=−µ+λδi1,(5.1.6)
with
µ=2
2Nd−1λ=Ndµ. (5.1.7)
The conditions for rule FII are expressed in terms of the new variables, by requiring
that Nd< n and x1< ph. As usual FIII follows FII, and this combination detaches
a high field domain from the emitter and creates a new one. The conditions for the
collision rule FIV is rephrased by requiring that one of the xibecomes zero.
We can then summarize this model by the following set of rules:
TI The high field lengths xievolve according to (5.1.5) until one of the following
rules applies.
TII If Nd< n and x1> phthen increase Ndby one, re-index xi→xi+1 for all i
and set x1= 0.
TIII If xi0= 0 then decrease Ndby one, re-index xi+1 →xifor all i≥i0.
In the following we will refer to the rules TI–TIII together with the initial con-
dition (5.1.4) as the tank model. The reason for this name will become obvious
in Sec. 5.2. The tank model has ncontinuous dynamical variables xi, i = 1 . . . n
and one discrete dynamical variable Nd. Like the front model (see page 49) it is
therefore a hybrid model. It furthermore depends on one discrete parameter n, and
the two continuous parameters phand Lh.
5.1.2 The Case pl>0
The above derivation of the tank model was restricted to the special case pl=
0. This restriction is not necessary for the derivation of the tank model, and we
now show that for general plthe rules TI–TIII are still valid without modification,
although the condition (5.1.1) and the definition of the time axis has to be adapted.
For pl>0 the front model rule FIII does not follow immediately FII, but the
injection of the depletion front is delayed, until a1> plis fulfilled. During this time
61
5 The Tank Model
λ
µ µ µ
1 2 n ph
xi
Figure 5.2: Scheme of an n-tank switched arrival system with minimal filling height
ph. The server filling rate is λ, the draining rate of all tanks is −µ.
we have Na=Ndand hence va=vd= 1, which means that no collisions occur,
and all front positions are just increased by pl. Equivalently, instead of adding the
constant plto every front position, one can also reduce the effective lattice length L
by the amount pleach time a new accumulation front is injected. During the transit
of an accumulation front to the emitter, this may happen at most L/(ph+pl) + 1
times, since two accumulation fronts are at least separated by a distance pl+ph.
Therefore the condition (5.1.1) that no fronts reach the collector has to be modified
for the case pl6= 0 to read
Lhn−1
2<3
4Lph
pl+ph−pl.(5.1.8)
Furthermore during the time between FII and FIII, all high field fronts are
bounded by an accumulation and a depletion front and ˙xi= 0 for all i. The net
effect of a non vanishing plis then to increase the time variable by the amount pl,
each time a high field front is disconnected from the emitter, but otherwise follow
the rules TI–TIII. This effect will obviously not influence the dynamic bifurcation
scenario, and can be eliminated completely by a suitable redefinition of the time
axis.
5.2 Connection to Water Tanks
Let us now justify the use of the term tank model for the model described by
the rules TI-TIII, by showing that surprisingly the same set of rules describes a
completely different system. Consider a system of nwater tanks as in Fig. 5.2.
Here a switching server fills one of the tanks with a filling rate λ, and at the same
62
5.3 The Poincar´e Map
time all Ndnonempty tanks drain at a rate −µ. To keep the total amount of water
at a constant value Lh, we require λ=µNd. The server switches to one of the n−Nd
empty tanks only under the condition that the tank which it is currently filling has
already reached the minimum filling height ph. This model is equivalent to what
we formulated by the rules TI-TIII and the initial condition (5.1.4). The variables
xiof the high field domains are up to some trivial re-indexing the filling heights
of the water tanks. The high field domain x1at the emitter is interpreted as the
tank connected to the server, while the other nonempty tanks represent detached
high field domains inside the superlattice. A switching of the server corresponds to
the detaching of the old high field domain at the emitter, and the generation of a
new one by TII. The rule that the server should not switch if the currently filled
tank has a filling height less than ph, obviously agrees with the requirement of TII
that a high field domain may only be detached from the emitter, if it has a certain
minimal length ph. The constant amount of water corresponds to the constant total
length of the high field regime Lh.
Variants of such models are well studied in the context of production processes
[89]. For example in [91] a model with a maximum filling height was considered. In
computer science similar models are relevant for the description of queuing systems
[94], where the server can for example represent a CPU, and the tanks are the differ-
ent tasks, which should be served by the CPU. Even the requirement of a minimal
filling height makes sense in this context, since in a multitasking computer system,
the switching of the task involves a certain overhead, which forbids arbitrarily fast
task switching.
5.3 The Poincar´e Map
A natural way to proceed is to consider a suitable Poincar´e section. Since all tanks
with the exception of tank #1, which is connected to the server, are equivalent, we
now adopt the sorting convention that xi> xi+1 for i≥2. Thus the dynamics of
the system is confined to an n−1 dimensional simplex of the form
An={x∈Rn|
n−i
X
j=1
xj=Lh∧x1≥0∧x2≥...≥xn≥0}.(5.3.1)
As a suitable hyperplane for the Poincar´e section we consider the n−2 dimensional
simplex
Bn={x∈An|x1≥ph∧xn= 0},(5.3.2)
which precisely contains the set of points, for which the conditions of rule TII are
fulfilled. A sketch of Anand Bnfor the case n= 3 is shown in Fig. 5.3.
Let us assume that at a certain time tmwe have x(tm)∈Bn. We now look for a
Poincar´e map
Pn:Bn→Bn,x(tm)7→ x(tm+1),(5.3.3)
63
5 The Tank Model
x1
x2
x3
A3
B3
ph
Figure 5.3: Sketch of the simplex A3(5.3.1) and the Poincar´e section B3(5.3.2) for
n= 3.
which relates x(tm) to the point x(tm+1) of the next visit of the simplex Bn. For
n= 2 the simplex B2is reduced to a point B2={(Lh,0)}, which by P2is simply
mapped onto itself. The dynamics is therefore trivially periodic. In the following
we assume n≥3.
The application of rule TII at tmtriggers the generation of a new high field domain
at the emitter, or in the language of water tanks, the switching of the server to a
new tank. This is achieved by a relabeling of the tank indices such that old x1is
enqueued among the x2. . . xn−1and the new x1is set to zero. Explicitly we write
x(t+
m) = MTIIx(tm),(5.3.4)
where t+
mdenotes the time just after the application of TII. The matrix MTII takes
care of the ordering of the filling heights and is given by
MTII =
δj0,j for j= 1,
δi,j for 2 ≤j < j0,
δi+1,j for j≥j0,
(5.3.5)
with xj0−1(tm)≥x1(tm)≥xj0(tm).(5.3.6)
In particular we note that
x1(t+
m) = 0,(5.3.7)
xn(t+
m) = min (xn−1(tm), x1(tm)) ,(5.3.8)
64
5.3 The Poincar´e Map
and therefore x(t+
m)/∈Bn. This guarantees that tm+1 > tm.
We first consider the case with (n−1)xn(t+
m)> ph, which by (5.3.8) is equivalent
to
(n−1)xn−1(tm)> ph.(5.3.9)
Then the tank #1 receives water from the n−1 other tanks, and will have reached
the filling height phbefore tank #nis empty. Therefore the time tm+1, at which
x(t) visits Bnis given by
tm+1 =tm+xn(t+
m)
µ.(5.3.10)
For t∈[t+
m, t−
m+1] there are no empty tanks, i.e Nd(t) = n, and we may write
explicitly
xi(tm+1) =
(n−1)xn(t+
m) for i= 1,
xi(t+
m)−xn(t+
m) for i= 2 . . . n −1,
0 for i=n.
(5.3.11)
In the case that (5.3.9) is not fulfilled, the last tank is empty before the first tank
has reached its minimal switching height ph. The switching time tm+1 is therefore
determined by the condition x1(tm+1) = ph. For the construction of the Poincar´e
map, we need to know the number of nonempty tanks ˜
Ndat the time t−
m+1 just
before we visit Bn. A little thought shows that this is given by
˜
Nd=Nd(t−
m+1) = max (k∈N
n
X
i=k+1
xi(t+
m) + (k−1)xk(t+
m)> ph)(5.3.12)
= max (k∈N
n−1
X
i=k
xi(tm) + (k−1)xk−1(tm)> ph).(5.3.13)
Using the definition
∆xe=ph−Pn−1
j=˜
Ndxj(tm)
˜
Nd−1,(5.3.14)
we find
tm+1 =tm+∆xe
µ,(5.3.15)
and finally
xi(tm+1) =
phfor i= 1,
xi(t+
m)−∆xefor i= 2 ... ˜
Nd,
0 for i > ˜
Nd.
(5.3.16)
Collecting the pieces together, (5.3.11), (5.3.16) and (5.3.4) define the Poincar´e
map Pnof (5.3.3) for general n. In the following we will explicitly examine the
cases P3and P4.
65
5 The Tank Model
In the limiting case of ph= 0 no tank has to wait for filling. We obtain a switched
arrival system [88] and the Poincar´e map can be written explicitly as
x(tm+1) = MTIIx(tm) + min [x1(tm), xn−1(tm)]
n−1
−1
.
.
.
−1
.(5.3.17)
As shown in [90] this system is chaotic for all n > 2 and has a constant invariant
probability measure.
5.4 Bifurcation Analysis for n= 3
In the case n= 3, the Poincar´e section B3in (5.3.2) is one-dimensional, and we
have for x∈B3the conditions x3= 0, x1∈[ph, Lh] and x2=Lh−x1(cf. Fig. 5.3).
Thus we may parametrize B3by the coordinate x1, and the Poincar´e map is fully
determined by a one-dimensional map
P3: [ph, Lh]→[ph, Lh], x1(tm)7→ x1(tm+1),(5.4.1)
which we will now determine explicitly.
Following (5.3.4) we find
x1(t+
m) = 0 (5.4.2)
x2(t+
m) = max [x1(tm), x2(tm)] = max [x1(tm), Lh−x1(tm)] (5.4.3)
x3(t+
m) = min [x1(tm), Lh−x1(tm)] (5.4.4)
and condition (5.3.9) can be written as
2 (Lh−x1(tm)) > ph.(5.4.5)
In the case that (5.4.5) is fulfilled we have from (5.3.11)
x1(tm+1) = 2 min [x1(tm), Lh−x1(tm)] ,(5.4.6)
and otherwise x1(tm+1) = ph.
Thus we may summarize the resulting Poincar´e map in the case n= 3 by
P3: [ph, Lh]→[ph, Lh] (5.4.7)
P3(x1) =
2x1for x1∈ph,1
2Lh
2Lh−2x1for x1∈1
2Lh, Lh−1
2ph
phfor x1∈Lh−1
2ph, Lh
(5.4.8)
= max {(Lh−|Lh−2x1|), ph}.(5.4.9)
The graph of this map is schematically drawn in Fig. 5.4(a) and for various values
of Lhin Fig. 5.4(b).
The dynamics of the iterated map (5.4.8) depends on the two positive1parameters
1Similar maps with negative phhave been considered in [95].
66
5.4 Bifurcation Analysis for n= 3
Lh
x1(tm)Lh
ph
ph
x1(tm+1)
a) b)
1 2 3
x1
1
2
3
P3(x1)
y = x
Lh = 3.0
Lh = 2.3
Lh = 1.6
Figure 5.4: (a) Schematic graph of the one-dimensional Poincar´e map P3for the
n= 3 tank model according to Eq. (5.4.8). In the shaded region the
map is not defined. (b) Graph of P3for ph= 1 and various values of
Lh.
phand Lh. The numerically calculated bifurcation diagram of P3(x1) for fixed ph
and increasing Lhis shown in Fig. 5.5. We see that we recover a bifurcation structure
which is very similar to the front model at low Lh(cf. Fig. 4.10). At any point
with the same Lh/phboth bifurcation diagrams show the same periodicity. This
is not surprising, since the only necessary condition in the derivation of the tank
model was that no fronts should reach the collector [see (5.1.1)]. The nature of the
bifurcations was not affected. However, the meaning of the variables has changed.
While in Fig. 4.10 the positions of the collisions is plotted, Fig. 5.5 shows the size
of the high field domain, when it is detached from the emitter. The information
about the position of the collisions was lost in the derivation of the tank model,
when the number of system variables was reduced from 2n−1 to n.
5.4.1 Connection with the Flat-Topped map
One-dimensional iterated maps are usually defined on the unit interval [0,1]. This
requirement may be met by an expansion of the domain of P3to [0, Lh] followed by
a rescaling off all lengths in units of Lh:
P3(x1) = Lhˆ
P3
ph
Lhx1
Lh,(5.4.10)
ˆ
P3
z(x) =
2xfor x∈0,1
2,
2−2xfor x∈1
2,1−1
2z,
zfor x∈1−1
2z, 1.
(5.4.11)
67
5 The Tank Model
1 2 3 4 5
Lh
1
2
3
4
x1
(P3)(i)
Lh
Lh -1/2
Figure 5.5: Bifurcation diagram of the Poincar´e map P3according to (5.4.8) for
fixed ph= 1 and varying Lh. Starting from a random x0
1∈[ph, Lh] we
calculate at each Lhthe ith iteration xi
1=P3(xi−1
1). The plotted points
are x200
1. . . x300
1. The blue and orange lines denote the left and right
boundaries of the flat region of P3, respectively.
0 1
1
0
λ
If
λ
0 1
1
0
z
IP
z
ˆ
P3
z(x)
fλ(x)
xx
a) b)
Figure 5.6: Graphs of (a) ˆ
P3
zaccording to Eq. (5.4.11) and (b) fλ(x) according to
Eq. (5.4.12).
68
5.4 Bifurcation Analysis for n= 3
The flat segment of the map ˆ
P3
z(x) is located at the right edge of its domain in
the interval IP
z= [1 −z/2,1] [cf. Fig. 5.6(a)]. In the mathematical and physical
literature, however, a slightly different class of flat-topped or trapezoidal maps of the
form [see Fig. 5.6(b)]
fλ(x) = min [1 −|2x−1|, λ] for λ∈[0,1] (5.4.12)
has been studied extensively [96, 97, 98]. The bifurcation diagram for this map is
shown in Fig. 5.7(a). Here the flat segment is at the maximum of the map in the
interval
If
λ= [λ/2,1−λ/2] = 1
2−1−λ
2,1
2+1−λ
2.(5.4.13)
The boundaries of If
λare indicated by colored lines in Fig. 5.7(a).
We observe that by choosing
λ= 1 −z
2= 1 −ph
2Lh
(5.4.14)
the flat segment of fhis exactly the preimage of the flat segment of ˆ
P3
z, i.e.
If
λ=ˆ
P3
z−1IP
z.(5.4.15)
Consider now the two trajectories x1, x2, x3,... and y1, y2, y3,... of some initial
point x0=y0/∈IP
z, with xi=ˆ
P3
z(xi−1) and yi=fλ(yi−1). Let mbe the first
index, such that xm∈IP
z. From (5.4.15) we conclude that the first index nfor
which xn∈If
λ, is given by n=m−1. Therefore the two trajectories yiand xi
are identical for i < m. Since ym−1∈If
λwe have ym=λand, using (5.4.14) we
find ym+1 = 2 −2λ=z=xm+1. This means that the trajectories x1, x2, x3, . . . and
y1, y2, y3,... only differ at indices mwith xm∈IP
z, where we have ym=λ. Apart
from this difference, all other properties of the two trajectories such as periodicity
or stability are identical. Hence we may restrict ourselves to the consideration of
the unimodal map fλ, which completely reproduces the bifurcation scenario of P3.
This equivalence can also be seen directly from the bifurcation diagram of P3
in Fig. 5.5. For any Lhthere is only one point in the interval between the blue
and the orange line. If we map this point to the blue line, we obtain exactly the
appropriately scaled bifurcation diagram of fλin Fig. 5.7(b).
5.4.2 The Tent-Map Case λ= 1
For λ= 1 [i.e. Lh→ ∞ according to Eq. (5.4.14)], the function f1(x) in Eq. (5.4.11)
reduces to the well known tent-map
f1(x) = 1 −|2x−1|.(5.4.16)
69
5 The Tank Model
a) b)
0.5 0.6 0.7 0.8 0.9 1
λ
0
0.2
0.4
0.6
0.8
1
x
fλ
(i)
λ/2
1-λ/2
11.5 22.5 33.5 44.5 5
1/z = 1/(2-2λ)
0.5
1
1.5
2
2.5
3
3.5
4
4.5
x/z = x/(2-2λ)
fλ
(i)
λ/2
1-λ/2
Figure 5.7: Bifurcation diagrams of the flat topped map fλ(x) [cf. Eq. (5.4.12) and
Fig. 5.6(b)]. The red and green lines show the left and right boundaries
of If
λ(5.4.13). Note that the left and right panels only differ in the axes
scaling.
0 0.2 0.4 0.6 0.8 1
x
0
0.2
0.4
0.6
0.8
1
f0
(k)(x)
k
5
4
3
2
1
p(4)
l
n(4)
l
Figure 5.8: Iterations of the tent-map f(k)
0for various kaccording to Eq. (5.4.17),
and fixed points p(k)
land n(k)
laccording to Eq. (5.4.18).
70
5.4 Bifurcation Analysis for n= 3
The tent-map is an archetype of a chaotic map [99] that can be treated analytically.
The results for this special case turn out to be useful in the discussion of the more
complicated case λ < 1 (see Sec. 5.4.3).
The kth iterate of f1, which we denote by f(k)
1has 2kbranches and is given by
f(k)
1(x) = 1 −2kx−2l+ 1
2kfor x∈l
2k−1,l+ 1
2k−1,l= 0 ...2k−1−1.(5.4.17)
The fixed points of f(k)
1, which are the points of period k, are explicitly given by
p(k)
l=2l
2k−1
n(k)
l=2l+ 2
2k+ 1
for l= 0 ...2k−1−1.(5.4.18)
The slopes of f(k)
1(x) at the fixed points are given by ∂xf(k)
1(p(k)) = 2kand ∂xf(k)
1(n(k)) =
−2k. Thus all fixed points are unstable, which means that the tent map f1has no
stable periodic orbits. The dynamics is chaotic [100] and has a constant invariant
measure2. Furthermore it follows that
f(k)
1(x)> x for x∈[p(k)
l, n(k)
l], l = 0 ...2k−1−1.(5.4.19)
In Fig. 5.8 the iterates f(k)
1, and the fixed points p(4)
land n(4)
lfor k= 4 are depicted.
It is worthwhile to note that the fixed points x(k)
lfollow a remarkable pattern,
when written in binary notation. The variable lin (5.4.18) can be written as a
binary number l= %Q, where Qis a string consisting of k−1 letters of 0 or 1
(we fill up with leading 0s as necessary) and the % indicates a binary number. We
denote by ˜
Qthe bitwise inverse of Q(i.e. % ˜
Q= 2k−%Q). A few lines of algebra
show that the fixed points in (5.4.18) are given in the binary number base by
p(k)
l=pk
Q= %0.Q0Q0Q0Q0Q . . .
n(k)
l=nk
Q= %0.Q1˜
Q0Q1˜
Q0Q . . . for l= %Q= 0 ...2k−1−1.(5.4.20)
The appearance of the patterns in (5.4.20) is also directly explained by considering
the tent map (5.4.16) in binary notation [99],
f1(%0.X) = (%0.Y for X= 0Y
%0.˜
Yfor X= 1Y . (5.4.21)
For f(k)
1we then find
f(k)
1(%0.X) = (%0.Y for X=Q0Y
%0.˜
Yfor X=Q1Y . for %Q= 0 ...2k−1−1 (5.4.22)
2Formally, there are infinitely many fixed points of the Perron Frobenius operator for f1, but
only the constant measure is natural, in the sense that it is stable against fluctuations (see
Exercise 7.5 in [99]).
71
5 The Tank Model
and requiring X=Yor X=˜
Yyields directly the patterns for pk
Qor nk
Qof
Eq. (5.4.18), respectively.
5.4.3 The Case λ < 1
In order to finally explain the bifurcation scenarios in Fig. 5.5, we now want to
characterize the stable periodic trajectories of fλ. In the following we will discuss
the trajectory of x0
λ= 1/2 given by
x1
λ, x2
λ, x3
λ,... with xk
λ=fλxk−1
λand x1
λ=fλ1
2.(5.4.23)
From (5.4.13) we see that x0
λ= 1/2∈If
λand thus x1
λ=λ. Let k=k(λ) be the
first index with xk
λ∈If
λand let us assume that3k < ∞. Then xk+1
λ=λand the
trajectory (5.4.23) has period k(λ). Since ∂xf(xk) = 0 we find
∂f(k)
λ(xi
λ)
∂x =
i+k−1
Y
j=i
∂fλ(xj
λ)
∂x = 0,for i≤k(5.4.24)
and the trajectory x1
λ,...,xk
λis a stable period k(λ) orbit.
We now want to determine the function k(λ). This can in principle be done,
by considering the iterates f(j)
λ, but this approach is analytically quite involved.
Instead, we make use of the known iterates f(j)
1of the tent map [see Eq. (5.4.17)].
Since fλdiffers from the tent map f1only in the interval If
λ, and xi
λ/∈If
λfor
1< i < k, we may write
xi
λ=f(i−1)
1(λ),for 1 < i ≤k. (5.4.25)
The condition xk
λ∈If
λfor a stable period korbit, can then be rephrased in term of
the tent map as f1(xk
λ)≥fλ(xk
λ) = λ. Formally we may thus express k(λ) as
k(λ) = min ni∈Nf(i)
1(λ)≥λo.(5.4.26)
This formula allows for a simple “graphical” interpretation with the help of Fig. 5.8.
To find k(λ), choose the point (λ, λ) on the diagonal, and find the smallest k, such
that f(k)
1is above the diagonal. In this way, we may for instance find k(λ) = 4 for
λ∈[p(4)
6, n(4)
6]. In the following we will show that all intervals with fixed kare of
this form.
Let us now consider the trajectories of xi
λunder variation of λ. Applying the
chain rule to (5.4.25) yields
∂xi
λ
∂λ = (−1)NR(i)2[(i−1) mod k],(5.4.27)
where NR(i) = jxj
λ>1
2∧1≤j≤[(i−1) mod k],(5.4.28)
3It was shown in [96] that the set {λ|k(λ)→ ∞} has Lebesgue measure zero.
72
5.4 Bifurcation Analysis for n= 3
and |·|denotes the cardinal number. Here NRcounts the number of minus signs that
are picked up by visiting the negative slope region of f1. The bifurcation parameter
λonly enters implicitly in the right hand side of Eq. (5.4.27) via k(λ). Let us for
example consider a λrange, for which a minimal k0exists, such that k0≤k(λ).
Then we have xi
λ/∈If
λfor i < k0, and NR(i) in (5.4.28) will be constant across the
considered λrange. Thus xi
λfor i≤k0will depend linearly on λby Eq. (5.4.27).
This naturally explains the appearance of the straight lines in Fig. 5.7(a) even
across complicated bifurcations. These straight lines are preserved under the axis
transformation leading to Fig. 5.7(b). We can now also explain the appearance of
the cobweb structures, for instance at λc= 5/6 [cf. 1/z = 3 in Fig. 5.7(b)]. This
yields a trajectory with xi
λ= 2/3 for i≥2. Thus k(λc) formally diverges, and we
can find intervals around λcwith arbitrary high k0. Therefore the points xk
λfor
2≥k≤k0will converge in straight lines to xk
λ→2/3 for λ→λc. This explains
the typical cobweb structure, where bundeles of straight lines appear to converge
in a single point.
Due to (5.4.27), the point xk
λ∈If
λhas the largest absolute slope with respect to
λof all points in the trajectory x0
λ,...,xk
λ. Bifurcations, i.e. a change in k(λ), will
only appear, if either with increasing λthe point xk
λleaves If
λ, or another point xi
λ
with i < k enters If
λ. This latter case is not independent from the first one, since
for any λ, there cannot exist simultaneously two distinct points xi
λ, xk
λin If
λ. Since
xk
λmoves continuously, it must leave If
λas xi
λenters it. With the help of (5.4.26)
and (5.4.19) we infer that the bifurcation points are fixed points of f(k)
1, and that
the intervals with constant k(λ) are of the form [see (5.4.20)]
Ik
Q=pk
Q, nk
Q,(5.4.29)
with a suitable binary string Qof length k−1. Suitable in this context means that
f(i)
1(λ)≤λfor all λ∈Ik
Qand i < k. (5.4.30)
The question is now, how to construct those suitable Q. The following con-
struction is essentially analogous to the classical construction of the universal U-
sequences [101] and will finally result in Table 5.1, where all Qs up to period 7 are
listed. Here we have the advantage that in our case all intervals can be calculated
explicitly, and we can avoid symbolic dynamics in the derivation, but in hindsight
we see that symbolic arguments yield essentially the same results. We stress that
the U-sequence is different from the well known Sarkovskii ordering [102, 100], since
the latter is only a statement about the existence of periods, and not about their
stability. The U-sequence however predicts the exact sequence of all stable periods,
as one bifurcation parameter is changed.
73
5 The Tank Model
#k Q pk
Qnk
Qitinerary
1 1 empty 0 0.10 empty
2 2 1 0.10 0.1100 R
3 4 110 0.1100 0.11010010 RLR
4 8 1101001 0.11010010 0.1101001100101100 RLR3LR
5 6 11010 0.110100 0.110101001010 RLR3
6 7 110101 0.1101010 0.11010110010100 RLR4
7 5 1101 0.11010 0.1101100100 RLR2
8 7 110110 0.110110 0.110111001000 RLR2LR
9 3 11 0.110 0.111000 RL
10 6 11100 0.111000 0.111001000110 RL2RL
11 7 111001 0.1110010 0.11100110001100 RL2RLR
12 5 1110 0.11100 0.1110100010 RL2R
13 7 111010 0.1110100 0.11101010001010 RL2R3
14 6 11101 0.111010 0.111011000100 RL2R2
15 7 111011 0.1110110 0.11101110001000 RL2R2L
16 4 111 0.1110 0.11110000 RL2
17 7 111100 0.1111000 0.11110010000110 RL3RL
18 6 11110 0.111100 0.111101000010 RL3R
19 7 111101 0.1111010 0.11110110000100 RL3R2
20 5 1111 0.11110 0.1111100000 RL3
21 7 111110 0.1111100 0.11111010000010 RL4R
22 6 11111 0.111110 0.111111000000 RL4
23 7 111111 0.1111110 0.1111110000000 RL5
Table 5.1: Intervals with constant period up to period 7 (with the exception of the
period 8 pattern in line 4).
5.4.4 Elementary Intervals
The most elementary strings Q, which fulfill the condition (5.4.30) are simply of
the form
Qk= 1 ...1
|{z}
k−1
= 1k−1,(5.4.31)
where we have used a convenient exponential notation ab, i.e. a b-fold repetition of
the letter a. Then we have from (5.4.29),
Ik
Qk= [%0.1k−101k−10...,%0.1k0k1k0k...],(5.4.32)
which means that any λ∈Ik
Qkis of the form λ= %0.1k−1X, with %0.˜
X≤λ.
Consequently, by
f(i)
1(λ) = 0.0k−i−1˜
X≤λfor 1 ≤i < k, (5.4.33)
74
5.4 Bifurcation Analysis for n= 3
condition (5.4.30) is fulfilled. On the other hand fk
1(λ)≥λby construction
[cf. (5.4.19)] and thus for λ∈Ik
Qk, we have indeed a stable period korbit. This
construction yields the lines #1,2,9,16,20,22,23 of Table 5.1.
5.4.5 Period Doubling Cascade
The next basic bifurcation scenario is the period doubling of any given suitable
pattern Q1. Assume that Q1fulfills (5.4.30) and consider the interval I2k
Q2with Q2
being the harmonic extension of Q1defined by
Q2=H(Q1) = Q11˜
Q1(5.4.34)
Then the boundaries of I2k
Q2are of the form
p2k
Q2= %0.Q20Q20Q20...= %0.Q11˜
Q10Q11˜
Q10Q11˜
Q10 = nk
Q1,(5.4.35)
n2k
Q2= %0.Q21˜
Q20Q21˜
Q20...= %0.Q11˜
Q11˜
Q10Q10Q11˜
Q11˜
Q10Q10. . . (5.4.36)
The interval I2k
Q2therefore connects consecutively to Ik
Q1from the right, with only
the boundary point in common.
We now want to show that I2k
Q2fulfills the condition (5.4.30). Assume that Q2
would not fulfill (5.4.30), i.e. we can find a λ∈I2k
Q2and i < 2k, such that f(i)
1(λi)>
λi. Since f(i)
1(nk
Q1)≤nk
Q1we find by continuity λ0∈Ik
Q1with f(i)
1(λf) = λf. This
λfis then also a fixed point of f(2i)
1, f(3i)
1,.... In particular, we may choose j=mi,
such that k≤j < 2k, and will find a
λj∈Ik
Q1with f(j)
1(λj)> λjand k≤j < 2k. (5.4.37)
Since
f(k)
1(p2k
Q2) = nk
Q1∈Ik
Q1(5.4.38)
f(k)
1(n2k
Q2) = %0.Q10Q11˜
Q1...∈Ik
Q1(5.4.39)
we have
f(k)
1(I2k
Q2)⊂Ik
Q1,(5.4.40)
and in particular f(k)(λj)∈Ik
Q1. By (5.4.37) we infer that f(j−k)(f(k)(λj)) > nk
Q1,
but this is not possible, since it would contradict the assumption that Q1fulfills
the condition (5.4.30). Thus Q2must also fulfill the condition (5.4.30), and I2k
Q2is
a suitable interval. This argument can be repeated for Q3=H(Q2), etc. leading to
a classical period doubling cascade.
We can now apply the period doubling construction to all patterns Qkfound in
Sec. 5.4.4. This yields the lines #2,3,4,10 of Table 5.1.
The first period doubling starting with the empty string Q1=Q1was also studied
by different methods in Ref. [98]. It was found that the Feigenbaum parameter δ,
75
5 The Tank Model
which is the ratio of two subsequent intervals in the period doubling cascade is not
constant but scales as
δ(k) = 2k.(5.4.41)
Since kitself obviously doubles at every period doubling, this yields an exponentially
fast convergence of the sequence of bifurcation points.
5.4.6 Intermediate Intervals
We now want to recursively construct the remaining Qs of Table 5.1. Assume that
we are given an ordered list of all intervals Iki
Qiup to a certain period kmax and let QA
and QBbe two strings characterizing two neighboring intervals IkA
QAand IkB
QBwith
nkA
QA< pkB
QB(which implies that QB6=H(QA)). Let us then consider the following
common substring Q, given by
nkA
QA= %0.Q0XA,(5.4.42)
pkB
QB= %0.Q1XB.(5.4.43)
We then see immediately that pk
Q< pkB
QBand nk
Q> nkA
QA. Thus the interval Ik
Qis
between the intervals IkA
QAand IkB
QB, but since we assumed that we had started with
a complete list up to period kmax, it follows that k > kmax, and Qis a suitable
string in the sense of (5.4.30). Applying this construction repeatedly to all pairs of
neighboring intervals, we can construct new lists with larger and larger kmax. This
finally yields all remaining lines in Table 5.1. With this construction we have thus
explicitly calculated the bifurcation points of the map fλ, and at the same time
solved the original bifurcation problem of P3. We know now the exact sequence of
periodic orbits as λ, or in the case of P3, the parameter Lhincreases. Up to about
period seven this sequence can be readily confirmed by the microscopic model (cf.
Fig. 4.6).
5.4.7 Symbolic Dynamics
At this point we seize the opportunity and make contact to the subject of symbolic
dynamics, which since the classical work of Metropolis, Stein and Stein [101] has
developed into a powerful tool in the study of universal features in nonlinear systems
[99, 96, 103, 104].
Let us consider the trajectory xi
λof the tent map and write a string, Mλ=
X1X2X3... with letters
Xi=(Lfor xi<1/2,
Rfor xi≥1/2. (5.4.44)
Instead of dealing with the explicit binary representation of λ, one can now use
the itinerary Mλinstead, since it can be shown that there exists a one to one
76
5.4 Bifurcation Analysis for n= 3
0246
Lh / ph
0
2
4
6
8
x1 / ph
2.75 3 3.25 3.5
1
2
3
Figure 5.9: Bifurcation diagram as in Fig. 5.5, but for the map (5.4.8) with a flat
region modified by a finite slope m= 0.001 (cf. Fig.5.6).
correspondence between the two representations [99]. The advantage of the itinerary
approach is that it is applicable to a large class of one dimensional maps, although
the construction of Table 5.1 is less explicit and intuitive [101]. Comparing the
itineraries in Table 5.1 with the Table in the appendix of Ref. [101], however shows
that both approaches are indeed equivalent.
5.4.8 Chaoticity
The maps P3and fλwe have considered in the previous sections show rich bifur-
cation scenarios, with infinitely long periods, which we can now explain sufficiently
well by means of the U-sequences. Nevertheless they are not truly chaotic. The rea-
son is obviously the flat segment, which will eventually be visited by the trajectory,
and will render any orbit stable. Such flat segments are however not physical, since
they would correspond to an exact projection of a continuous set of phase points
onto one single phase point. Since in the derivation of the tank model, a number of
approximations were made, it is more likely that the flat segment is not exactly flat,
but has at least a tiny slope 0 < m 1. Since this finite slope however increases in
the iterated map as 2kmit will destabilize periodic orbits of period k(λ)>log(1/m)
and result in chaotic behavior. As shown in Fig. 5.9, this leads to continuous bands
similar to the ones observed in the microscopic model (cf. Fig. 4.6) .
77
5 The Tank Model
5.5 Bifurcation Analysis for n= 4
In the case n= 4, the Poincar´e simplex B4in (5.3.2) is two-dimensional, and may
be conveniently parametrized by x1and x3as follows:
B4=x∈R4|x1∈[ph, Lh]∧x3∈[0,(Lh−x1)/2] ∧x2=Lh−x3−x2∧x4= 0.
(5.5.1)
The corresponding region in the (x1, x3) plane is shown in Fig. 5.10(a). We may
now construct the associated two-dimensional Poincar´e map,
P4:
B4→B4
x1(tm)
x3(tm)7→ x1(tm+1)
x3(tm+1).(5.5.2)
We first note that according to Eq. (5.3.13) the domain Bassumes the following
partition with respect to ˜
Nd,
˜
Nd= 4 ⇔x3(tm)>ph
3(5.5.3)
˜
Nd= 3 ⇔x3(tm)>2Lh−2x1(tm)−ph(5.5.4)
˜
Nd= 2 else,(5.5.5)
which is indicated by the color scheme in Fig. 5.10(a). For ˜
Nd= 4 we have
x1(tm+1) = 3 min [x1(tm), x3(tm)] (5.5.6)
x3(tm+1) = |min [x1(tm), Lh−x1(tm)−x3(tm)] −x3(tm)|.(5.5.7)
The resulting three branches are marked by three different red colors in Fig. 5.10(a).
The ranges of these three branches partly overlap, as is indicated by the striped
regions in Fig. 5.10(b). In particular points with x1∈(3ph, Lh) have in general
three preimages, one from each branch. In the case of ˜
Nd= 3, we find for P4:
x1(tm+1) = ph(5.5.8)
x3(tm+1) = min [x1(tm), Lh−x1(tm)−x3(tm)] −ph−x3(tm)
2.(5.5.9)
These two branches correspond to the two green colored regions in Fig. 5.10(a),
which are mapped to the green lines in Fig. 5.10(b). Finally for ˜
Nd= 2 the Poincar´e
map P4reduces to
x1(tm+1) = ph(5.5.10)
x3(tm+1) = 0.(5.5.11)
This corresponds to the yellow region in Fig. 5.10(a), which is mapped to the yellow
dot in in Fig. 5.10(b). The Jacobian of (5.5.10) vanishes, and therefore any periodic
trajectory visiting the ˜
Nd= 2 region will be stable.
78
5.5 Bifurcation Analysis for n= 4
Lh
ph
x3
Lh
Lh
2
ph
3
ph
Lh-ph
2
Lh
2
0
Lh
x1
ph
Lh
Lh
2
ph
0
3ph
ph
2
x3
b)
a) ph
2
Lh
2-
Figure 5.10: Graphical representation of the Poincare map P4. (a) The colored areas
define the domain B4of the Poincar´e map P4(5.5.2) in the (x1, x3)
parametrization. The reddish areas indicate the domain of ˜
Nd= 4,
greenish colors indicate ˜
Nd= 3 and the yellow color the ˜
Nd= 2 regions
(cf. Eq. (5.3.13). (b) range of the colored regions from (a) under P4.
The colors in (b) correspond to the color of the preimages in (a). The
striped areas and striped lines, indicate regions with more than one
preimage. The yellow region in (a) is mapped to the yellow dot in (b)
at (ph,0).
79
5 The Tank Model
2 3 4 56
Lh
1
2
3
4
x1
Figure 5.11: Bifurcation diagram of x1versus Lhfor the Poincar´e map P4given by
(5.5.12). Red, green, and yellow dots fall into regions with ˜
Nd= 4, 3,
and 2, respectively (cf. Fig. 5.10).
Combining the three Eqs. (5.5.6) (5.5.8) (5.5.10) we can express the Poincar´e
map explicitly by
P4:B4→B4,x(tm)7→ x(tm+1) (5.5.12)
x1(tm+1) = max {3 min [x1(tm), x3(tm)] , ph}(5.5.13)
x3(tm+1) = max (0, x3(tm)−x1(tm),(5.5.14)
min [x1(tm), Lh−x1(tm)−x3(tm)] −max x3(tm),ph−x3(tm)
2).
P4is obviously continuous, since max and min are continuous.
The bifurcation diagram for P4is shown in Fig. 5.11. Note that the bifurcation
scenario is now remarkably different from the n= 3 case (Fig. 5.5) but instead
resembles the corresponding diagram of the front model for n= 4 [see Fig. 4.11(a)].
In particular the cobweb structure is now less pronounced, than in the n= 3
case, and the large period three window is missing. The general theory for two
dimensional iterated maps of the type P4is considerably more involved than for the
one dimensional case. A systematic approach in the language of border collisions
was proposed in [105, 106] (for a practical application see also [107]). Since we
did not see a clear evidence for n= 4 behavior from the microscopic superlattice
model, we are not pursuing this path further, although this bifurcation scenario is
of fundamental interest for the tank model.
80
6 Nonstationary External Voltage
So far we have only considered configurations with a fixed external voltage U.
However recent experimental [108, 109, 47, 110, 50] and theoretical [68, 111, 112, 49]
results show that qualitatively new features occur under nonstationary external
voltage conditions.
In this chapter we will first consider switching processes, where the external
voltage is increased instantaneously at a certain time t= 0, and then proceed to
consider ramping processes, where the external voltage is continuously increased
over one or more discontinuities of the current voltage characteristic. Parts of those
theoretical considerations have been published in Refs. [68, 111] and were later
confirmed experimentally with astonishingly high accuracy by Rogozia et al. [110].
At the end of this chapter we will also briefly examine the behavior of a super-
lattice under an external voltage, which is the sum of an ac and dc voltage. Such
a configuration has been experimentally considered in the case of driven chaos [80]
and high frequency oscillations in a resonator [50], but here we are mainly interested
in theoretical predictions for the frequency dependent impedance of the superlattice
device.
For easier comparison with existing work, we now use a superlattice of type
A (cf. Table 2.2 on page 12), with N= 40 wells and a cross section of A=
14400 µm2, at a temperature of T= 5 K. These parameters are in accordance with
the experimental superlattice used in the switching and sweeping experiments in
Refs. [109, 47, 110] and were also used as the starting point for the theoretical
consideration in [68, 111, 112]. We use again simple Ohmic boundary currents at
the emitter and the collector (cf. Sec. 2.3). Reasonable agreement with the overall
shape of the experimental current–voltage characteristic is obtained by choosing the
contact conductivity σ= 0.01 (Ωm)−1.
From the homogeneous current density characteristic in Fig. 6.1(a) we note that
the critical current density jcis larger than the maximum current density js
max for
which stationary accumulation fronts occur (cf. velocity current characteristic in
Fig. 6.1(b)). Then according to (3.3.5) the low field domain at the emitter is sta-
ble, and the complete current-voltage characteristic exhibits the typical sawtooth
pattern of Fig. 6.2 [55, 56, 113, 114]. The upper and lower branches in Fig. 6.2
correspond to the up- and down-sweep of the external voltage, respectively. Each
upper branch can be continuously extended to a lower branch, by slowly decreasing
the voltage (violet lines in Fig. 6.2). Then each branch corresponds to a configu-
ration with a stationary electron accumulation front located at one particular well.
At the discontinuity points of a branch, the accumulation front moves to a different
81
6 Nonstationary External Voltage
-10
-5
0electric field [MV/m]
-0.04
-0.03
-0.02
-0.01
0
current density [A/mm2]
(Fc, jc)
js
max
js
min
σ = 0.1 Ω−1m-1
σ = 0.01 Ω−1m-1
σ = 0.001 Ω−1m-1
0 0.002 0.004 0.006 0.008 0.01 0.012
current density [A/mm2]
-20
-10
0
10
20
30
40
velocity [wells/µs]
js
min js
max
jdj(2,1)
depletion front
accumulation front
a) b)
Figure 6.1: (a) Current density vs electric field characteristic between neutral wells
(black) and at the emitter, with various conductivities σfor superlattice
A (Table 2.2). (Fc, jc) denotes the intersection point of the two char-
acteristics. The shaded area between the current values js
min and js
max
marks the regime of stationary accumulation fronts. (b) Corresponding
velocity current characteristic.
well, and thus the operating point jumps onto a new branch. For one value of the
external voltage, we find in general multiple stable solutions which are associated
with different branches and different currents. For the parameters of superlattice
A, we observe threefold (cf. U= 1.5 V) and fourfold (cf. U= 1.8 V) multistability.
6.1 Switching
Let us first consider the situation, where the time-dependent external voltage is
given by a step function of the form
U(t) = (Uifor t < 0,
Uf=Ui+Ustep for t≥0, (6.1.1)
with the step size Ustep and the initial and final voltage Uiand Uf, respectively. We
will use the terms up jump and down jump for the cases Ustep >0 and Ustep <0,
respectively.
We start from an operating point at t < 0, which is on the upper branch for
the fixed initial voltage Ui= 1.5 V, and apply voltage steps at t= 0 of various
sizes. The initial and final operating points on the current voltage characteristic
are denoted by arrows in Fig. 6.2.
The most simple scenario occurs if Ustep is small enough, such that the initial
branch is still present at the final voltage Uf=Ui+Ustep, as for Ustep =−0.2 V
82
6.1 Switching
1 1.2 1.4 1.6 1.8
voltage U [V]
0
50
100
150
current J [µΑ]
-0.5 -0.2 0 0.1 0.2 0.3
Ustep [V]
0.000
0.002
0.004
0.006
0.008
0.010
0.012
current density [A/mm2]
Ucrit
Figure 6.2: Up sweep (red) and down sweep (blue) current-voltage characteristic
for a stationary superlattice of type A(parameters as in Table 2.2),
σ= 0.01(Ωm)−1. The intermediate branches (violet) are obtained by
sweeping along each individual branch. The arrows denote the starting
and end points of various switching scenarios. For Ustep > Ucrit final
operating points are on the down-sweep branch.
in Fig. 6.2. Then the system prefers to remain on the initial branch (unless we are
very close to a discontinuity point, as we will see later). Since the maximum of the
electron distribution remains at the same well, no major charge redistributions are
involved and only the position of the center of charge pa(see (3.1.7)) is shifted to
account for the changed voltage. Thus the current response in Fig. 6.3 shows an
almost instantaneous relaxation to the final current value, which is in agreement
with the experimental data of [110] as shown by the current trace A in Fig. 6.4(a).
6.1.1 Down Jumps
If the initial branch does not exist at the final voltage, the system is forced on a new
branch. However, due to the multistability of the system, it is not a priori clear,
which branch will be chosen. For a down jump with Ustep <0 we observe the lower
branch at the final voltage is always preferred, independently of the size of Ustep.
Consider for instance the case Ui= 1.5 V and Ustep =−0.5 V. From Fig. 6.2 we see
that we will finally end up on the lowest stable branch for Uf= 1.0 V. The current
response for this case (violet line in Fig. 6.3) shows a sharp drop at the switching
time t= 0, which is due to the instantaneous decrease of all electric fields by Ustep/L.
The current density then drops below js
min, and the accumulation front will move
in positive direction, towards the collector, as shown in the electron density plot in
Fig. 6.5. As the accumulation front moves to the collector, the high field domain
83
6 Nonstationary External Voltage
-2 0 2 4 68 10
time [µs]
0
50
100
150
200
250
current J [µΑ]
-0.51 V
-0.5 V
-0.2 V
+0.1V
+0.25V
+0.26V
+0.4V
Figure 6.3: Current response versus time for Ui= 1.5 V and various Ustep. Parame-
ters as in Fig.6.2. The individual current traces are spread by 25 µA for
clarity.
a) b)
verse relocation time is expected according to Eq. ~3!to
exhibit a square-root dependence on the final voltage.
For jumps between next-nearest-neighboring branches,
the CAL jumps first to the adjacent well according to the
intermediate current level of curve Cin Fig. 4~a!, before it
moves with a stochastically varying delay time to its final
position. This intermediate current level is below Il, so that
it is located on the unstable part of this branch, where the
current decreases almost linearly with decreasing V1. There-
fore, we can also observe a square-root-like dependence of
1/
t
reloc as a function of the final voltage, as shown by the full
dots in Fig. 4~b!for down jumps between next-nearest-
neighboring branches. The situation can be generalized for
down jumps between branches, which are even further apart.
The CAL first moves quickly to the adjacent well of its final
location and then, after a stochastically varying delay time,7
jumps to its final position. We conclude that for down jumps
the relocation of the domain boundary can always be de-
scribed by a direct motion of the CAL alone in agreement
with the theoretical predictions.9,11
V. UP JUMPS: MONOPOLE AND TRIPOLE
RELOCATION PROCESSES
For up jumps, the CAL has to move against the flow of
the electrons. Figure 5~a!shows current traces for small up
jumps ~curves Aand B) starting from the third branch. The
peak current Ipis now larger than the initial value I0because
of the positive displacement current. Recently, we have in-
vestigated the probability distribution of the switching time
for small up jumps from the third branch to the beginning of
the fourth branch of the I-Vcharacteristic of this sample.7
The CAL jumps after a delay time against the electron flow.
For jumps to voltages far from the discontinuity, the distri-
bution function of the relocation time is narrow and Gauss-
ian. However, when the discontinuity is approached from
above by reducing the V1, the average relocation time in-
creases by more than one order of magnitude. At the same
time, the distribution function changes to a first-passage-time
distribution with a steep increase for short times and a broad
tail for long times.
For these small up jumps, the relocation process is very
similar to the one for down jumps. After the initial peak, the
current decreases slightly during a certain time interval until
a critical current level is reached. Then the current decreases
quickly to its final value. As for the down jumps, the inverse
relocation time as defined in Fig. 4~a!also exhibits a square-
root-like dependence on the final voltage V1for these small
up jumps ~not shown!. Figure 5~b!shows the peak Ipand
final current values I1compared with the I-Vcharacteristic.
The behavior is very similar to the one for the down jump
~cf. Fig. 3!, taking into account the different direction of the
voltage step.
For larger up jumps, the current shows a very different
behavior as indicated by curve Cin Fig. 5~a!. After the peak,
the current decreases rapidly to a range well below the stable
current region of the I-Vcharacteristic. There it fluctuates
around this value for a time period
t
dipole'2
m
s, after which
it rises to an intermediate value. It then remains for a sto-
chastically fluctuating delay time
t
dat this intermediate
level, before it switches to its final value. Figure 5~b!shows
that the final current level for larger up jumps is located on
the down-sweep characteristic.
Figure 6 shows the low-current region of the transients for
jumps from the third branch to the fifth branch ~curve A) and
from the tenth to the 13th branch ~curve B). After the initial
peak ~region 1!, a region of spikes with rather irregular am-
plitudes ~region 2!follows, which is terminated by a larger
spike. The duration of region 2 depends mainly on V1. After
FIG. 4. ~a!Typical real-time traces for jumps from the fourth to
the third (Aand B) branch and to the second branch ~C!to define
t
reloc .~b!Inverse relocation time as a function of V1for jumps
from the third to the second branch ~circles!and to the first branch
~dots!along with the I-Vcharacteristic ~squares!. The data in ~b!
are fitted with a square-root dependence on
u
V12Vth
u
, where Vth is
defined in the text.
FIG. 5. ~a!Real-time traces as well as ~b!peak Ipand final
current I1compared with the I-Vcharacteristics ~solid line!for up
jumps starting from the third branch. Curves A,B, and Ccorrespond
to final voltages V150.798, 0.846, and 0.853 V, respectively. The
short-dashed line in ~b!indicates the separation between the mono-
pole and tripole relocation process, and the long-dashed line is used
as a guide to the eye.
ROGOZIA, TEITSWORTH, GRAHN, AND PLOOG PHYSICAL REVIEW B 65 205303
205303-4
layers of the contacts, the actual voltage for a current jump
may differ between different measurements, in particular af-
ter heating the sample to room temperature and cooling it
again to low temperatures. Details of this effect have been
described in Ref. 13.
The two-dimensional charge density n2d at the domain
boundary can be calculated from Poisson’s equation
n2d5
ee
0
DF
e,~1!
where
e
and
e
0denote the dielectric constants of the material
and of the vacuum, respectively, DFthe electric field change
at the domain boundary, and ethe elementary charge. For a
field-strength difference of DF5120 mV/13 nm and an av-
erage dielectric constant
e
;12 for GaAs and AlAs, we ob-
tain a carrier density of n2d56.131011 cm22for a fully
developed CAL. The nominal doping density of the GaAs
wells corresponds to n2d51.531011 cm22, which is a about
a factor of 4 smaller than the carrier density necessary to
form the domain boundary within a single well. Since the
number of jumps in the I-Vcharacteristic is correlated with
the number of periods, the domain boundary in the static
case is formed by a CAL within a single well. In contrast, a
charge depletion layer ~CDL!would extend over at least four
wells.
IV. DOWN JUMPS: MONOPOLE RELOCATION
For down jumps, the negative CAL can move according
to the direction of the applied electric field. Therefore, we
always expect to observe a simple monopole relocation for
switching to smaller voltages. Figure 3~a!shows the current
transients for a voltage jump from an initial voltage V0on
the third branch to three lower values V1. In the first 8 ns, the
current decreases due to the displacement current3,6 (jdisp
5
ee
0dF/dt) from the initial value I0to a downward peak
value Ipgiven by
Ip5I01Idisp5I01
ee
0
A
L
dV
dt ,~2!
where Adenotes the area of the mesa and Lthe length
of the SL.
If V0and V1are on the same branch ~trace A), the final
current I1is reached after
t
p'30 ns. For jumps to other
branches, the current stays after
t
pfor some time at an al-
most constant level ~curves Band C), before it increases
during a short switching time to its final value (I1), which is
located on the down sweep characteristic. The stochastic as-
pects of the switching, resulting in different distribution
functions, depend on the final voltage separation from the
discontinuity and are described in detail in Ref. 7 for small
up jumps.
Figure 3~b!shows the current value of Ipand I1as a
function of the final voltage V1, together with the corre-
sponding up and down sweep of the I-Vcharacteristic. While
I1just follows the down sweep of the time-averaged I-V
characteristic, Ipdecreases linearly with decreasing V1inde-
pendently of the final current branch as described by Eq. ~2!.
For larger down jumps, Ipcan even become negative.
Amann et al.9and Carpio et al.11 calculated the velocity
of fully developed CAL’s separating a LFD on the emitter
and a HFD on the collector side as well as CDL’s separating
a HFD on the emitter and a LFD on the collector side as a
function of the current. In the current range of the stable
branches between a lower and upper critical current denoted
Iland Iu, respectively, the position of the CAL does not
change. For a constant current Iwith I,Il(I.Iu), the CAL
moves toward the collector ~emitter!, while for Il,I,Iuit
remains stationary. Near the critical current, the velocity
vaccu of the CAL is predicted to scale as9,11
vaccu}
A
u
I2Il/u
u
.~3!
However, for a CDL, the velocity vdepl is always positive,
i.e., it always moves toward the collector. In this case, vdepl
depends almost linearly on the current. Since we are looking
at down jumps (I,Il), the CAL can move directly towards
the collector. Therefore, the relocation of the domain bound-
ary for down jumps is expected to be a simple motion of the
CAL alone.
In Fig. 4~a!, typical time traces for down jumps between
two adjacent branches ~curves Aand B) and next-nearest-
neighboring branches ~curve C) are shown. After the dis-
placement current peak, the current remains for a certain
time at an intermediate level and then switches to its final
value. We define a relocation time
t
reloc as the time delay
between the initial voltage step and the switching to the final
current value. The inverse relocation time is proportional to
vaccu . For curves Aand C,
t
reloc is much larger than for curve
B, because V1is close to the respective discontinuity. In Fig.
4~b!, the inverse of
t
reloc is shown for a down jump between
adjacent branches ~open dots!. It has a square-root depen-
dence on
u
V12Vth
u
, where Vth denotes the voltage, at which
the respective current discontinuity occurs. Since the peak
current is a linear function of V12V0@cf. Eq. ~2!#, the in-
FIG. 3. ~a!Averaged time traces as well as ~b!peak Ipand final
current I1compared with the I-Vcharacteristics ~solid line!for
down jumps from the third branch to lower voltages V1as indi-
cated. The arrow in ~a!indicates the relocation time
t
reloc . The
dashed line in ~b!is used as a guide to the eye.
RELOCATION DYNAMICS OF DOMAIN BOUNDARIES IN . . . PHYSICAL REVIEW B 65 205303
205303-3
Figure 6.4: (a) Experimental time traces for down jumps from the third branch
to the third (A), second (B) and first branch (C), respectively. (b)
Experimental time traces for down jumps from the fourth branch to the
third branch (A and B) and second branch (C), respectively. The final
voltage for trace A is closer to the discontinuity with the fourth branch
than for current trace B. Note that the time scales of (a) and (b) are
different. Reprinted from [110].
84
6.1 Switching
Ustep = +0.4V
Ustep = +0.26V
Ustep = +0.25V
Ustep = -0.5V
time [µs]
0 10
well
Figure 6.5: Space time plot of the electron density evolution of the switching pro-
cesses indicated in Fig.6.3, with Ui= 1.5 V and various Ustep. Regions
of electron accumulation (depletion) are shaded in blue (red).
decreases, and thus the fields in the low and high field domain, will increase again.
This leads to an increase in the current, which is however intermitted by small
spikes, due to the motion of the maximum of the electron density to a new well.
Thus for Ustep <0 we have always a direct motion of the accumulation front to its
new position, irrespective of the size of Ustep. This finding is corroborated by recent
experiments [110] (cf. Fig. 6.4(b)).
A further remarkable feature in the case Ustep =−0.5 V is the plateau in the
current trace (Fig. 6.3) from t= 0.7...2.2µs, just before the last spike brings us
to the final current value. This is caused by the vicinity of Vfto the discontinuity
of the down-sweep characteristic (Fig. 6.2). As the accumulation front approaches
this discontinuity, the current density is only slightly below js
min, and the accumu-
lation front will accordingly move rather slowly, as is also evident from the electron
density plot in Fig. 6.5. This effect is further enhanced by the flat slope at the dis-
continuity points of the down-sweep branches (as opposed to the steep slope at the
discontinuity points of the up-sweep branches). If the final point is however farther
away from the discontinuity, the switching time is reduced as is demonstrated by
the current trace for Ustep =−0.51 V in Fig. 6.3). The sharp spikes, associated with
the well to well hopping of the maximum electron concentration can not be resolved
85
6 Nonstationary External Voltage
experimentally, however the plateau structure and the dependence of the switching
time on the distance from the discontinuity agrees with the experimental results as
shown by the experimental current traces in Fig. 6.4(b) [110].
6.1.2 Up Jumps
For up jumps, the final operating point turns out to depend non trivially on the
exact value of Ustep. Consider again the fixed initial voltage Ui= 1.5 V. Then there
exists a threshold voltage step Ucrit, such that for small positive jumps Ustep < Ucrit,
the final position of the operating point is located at the upper branch of the final
voltage Ufon the next or the next but one branch (cf. orange and dark green arrows
in Fig. 6.2). The corresponding current traces (Ustep = +0.1 V and Ustep = +0.25 V
in Fig. 6.3) show a sharp current peak at the switching time, due to the sudden
increase of all electric fields in the superlattice. As long as the current density
surpasses js
max the accumulation front acquires a negative velocity due to Fig. 6.1(b).
The front moves towards the emitter, as is evident from the charge density evolution
for Ustep = +0.25 V in Fig. 6.5. This motion decreases the fields again by Gauss’s
law (3.0.2), until the current density drops below js
max and the accumulation front
stops. Similar to the case of down jumps, the relaxation time τruntil the final
current is reached depends sensitively on the distance of the final operating point
from the previous branch discontinuity. Typically we have τr<0.5µs.
For Ustep > Ucrit, the system behavior changes dramatically. Now the final op-
erating point is on the lower branch of the final voltage Uf. This is surprising,
since points on the down sweep branches, can ordinarily only be reached after first
sweeping to higher voltages. Also the current trace shows a remarkable behavior,
as shown for the voltages Ustep = +0.26 ,+0.4 V in Fig. 6.3. The initial peak at the
switching time, is followed immediately by a sharp drop to a level well below js
minA.
The current then shows a spiky behavior, but remains on this plateau for about
6µs. It finally rises to a constant value, which corresponds to an operating point,
which is located on the lower branch of the current voltage characteristic.
The explanation for this puzzling scenario can be seen from the electron density
evolution in Fig. 6.5 (upper panels for Ustep = +0.26 ,+0.4 V). We see that at the
switching time t= 0, the original accumulation front moves towards the emitter,
just as in the Ustep = +0.25 V case. But at the same time, a new dipole, consisting of
a leading depletion and a trailing accumulation front is injected at the emitter. As
we have learned in Section 3.3.1, this dipole injection is triggered by the rise of the
current beyond jcfor a sufficiently long time. This triggering condition is fulfilled
for Ustep > Ucrit. After the injection of the dipole, we obtain a tripole configuration,
and according to Sec. 3.2, the current density is fixed at j(2,1) (Fig. 6.1), which
explains the plateau like current trace in Fig. 6.3, which is also visible in the exper-
imental data in Fig. 6.7(a). The well to well hopping of the fronts of the tripole are
responsible for the spikes in this plateau. As the fronts move towards the collector,
the depletion front moves at twice the velocity of the two accumulation fronts, as
86
6.1 Switching
2.2V
time [µs]
0 10
0.7V
a) b)
-2 0 2 4 68 10
time [µs]
0
50
100
150
current J [µΑ]
0
0.002
0.004
0.006
0.008
0.01
current density [A/mm2]
Ui = 2.2V
Ui = 1.5V
Ui = 0.7V
Ui
Figure 6.6: Current trace (a) and density evolution (b) for switching processes with
varying Uiand fixed Ustep = +0.3V.
discussed in Sec. 3.2, and reaches the collector shortly after the original electron
accumulation front. Meanwhile the new accumulation front has traveled from the
emitter towards the center and reaches its final position after the two other fronts
have disappeared from the sample. We can now explain, why the final operating
point is on the lower branch of the stationary current voltage characteristic. During
a down sweep, the accumulation front also moves towards the collector, driven by a
current below js
min. But since the same is true for the newly generated accumulation
front, both processes have to lead to the same final state.
For Ui= 1.5 V, the original accumulation front and the depletion front reach the
collector at almost the same time. By choosing different initial voltage Ui, we can
select other scenarios. Consider for instance the case Ui= 0.7 V in Fig. 6.6. Now
the original accumulation front quickly reaches the collector, and we are left with a
dipole. This is also visible from the current trace, which switches from the tripole
current j(2,1)Ato the lower dipole current jdA, during the dipole phase. Although
this effect seems to be small, it was demonstrated experimentally (cf. Fig. 6.7(b))
[110]. On the other hand for a larger Ui= 2.2V, the fast depletion front catches up
with the original accumulation front, before it reaches the collector. In this case,
the dipole phase is absent, and a shortened current trace is obtained (Fig. 6.6.)
The fact that dipole fronts are injected is related to our choice of the boundary
currents. In Refs. [73, 115] switch on processes were considered and it was found
that the domain formation is caused by a monopole mechanism. We can obtain
equivalent scenarios by choosing σ= 0.1 Ω−1m−1, as shown in Fig. 6.8. Now with
increasing Ustep, there is a continuous transition from a direct relocation of the
charge accumulation front (Ustep = +0.4 V) to a scenario, where the original ac-
cumulation front vanishes and a new accumulation front generated at the emitter
forms the new domain boundary (Ustep = +0.9 V). At an intermediate voltage step
of Ustep = +0.5 V, the original and the newly generated front at the emitter merge
87
6 Nonstationary External Voltage
a) b)
verse relocation time is expected according to Eq. ~3!to
exhibit a square-root dependence on the final voltage.
For jumps between next-nearest-neighboring branches,
the CAL jumps first to the adjacent well according to the
intermediate current level of curve Cin Fig. 4~a!, before it
moves with a stochastically varying delay time to its final
position. This intermediate current level is below Il, so that
it is located on the unstable part of this branch, where the
current decreases almost linearly with decreasing V1. There-
fore, we can also observe a square-root-like dependence of
1/
t
reloc as a function of the final voltage, as shown by the full
dots in Fig. 4~b!for down jumps between next-nearest-
neighboring branches. The situation can be generalized for
down jumps between branches, which are even further apart.
The CAL first moves quickly to the adjacent well of its final
location and then, after a stochastically varying delay time,7
jumps to its final position. We conclude that for down jumps
the relocation of the domain boundary can always be de-
scribed by a direct motion of the CAL alone in agreement
with the theoretical predictions.9,11
V. UP JUMPS: MONOPOLE AND TRIPOLE
RELOCATION PROCESSES
For up jumps, the CAL has to move against the flow of
the electrons. Figure 5~a!shows current traces for small up
jumps ~curves Aand B) starting from the third branch. The
peak current Ipis now larger than the initial value I0because
of the positive displacement current. Recently, we have in-
vestigated the probability distribution of the switching time
for small up jumps from the third branch to the beginning of
the fourth branch of the I-Vcharacteristic of this sample.7
The CAL jumps after a delay time against the electron flow.
For jumps to voltages far from the discontinuity, the distri-
bution function of the relocation time is narrow and Gauss-
ian. However, when the discontinuity is approached from
above by reducing the V1, the average relocation time in-
creases by more than one order of magnitude. At the same
time, the distribution function changes to a first-passage-time
distribution with a steep increase for short times and a broad
tail for long times.
For these small up jumps, the relocation process is very
similar to the one for down jumps. After the initial peak, the
current decreases slightly during a certain time interval until
a critical current level is reached. Then the current decreases
quickly to its final value. As for the down jumps, the inverse
relocation time as defined in Fig. 4~a!also exhibits a square-
root-like dependence on the final voltage V1for these small
up jumps ~not shown!. Figure 5~b!shows the peak Ipand
final current values I1compared with the I-Vcharacteristic.
The behavior is very similar to the one for the down jump
~cf. Fig. 3!, taking into account the different direction of the
voltage step.
For larger up jumps, the current shows a very different
behavior as indicated by curve Cin Fig. 5~a!. After the peak,
the current decreases rapidly to a range well below the stable
current region of the I-Vcharacteristic. There it fluctuates
around this value for a time period
t
dipole'2
m
s, after which
it rises to an intermediate value. It then remains for a sto-
chastically fluctuating delay time
t
dat this intermediate
level, before it switches to its final value. Figure 5~b!shows
that the final current level for larger up jumps is located on
the down-sweep characteristic.
Figure 6 shows the low-current region of the transients for
jumps from the third branch to the fifth branch ~curve A) and
from the tenth to the 13th branch ~curve B). After the initial
peak ~region 1!, a region of spikes with rather irregular am-
plitudes ~region 2!follows, which is terminated by a larger
spike. The duration of region 2 depends mainly on V1. After
FIG. 4. ~a!Typical real-time traces for jumps from the fourth to
the third (Aand B) branch and to the second branch ~C!to define
t
reloc .~b!Inverse relocation time as a function of V1for jumps
from the third to the second branch ~circles!and to the first branch
~dots!along with the I-Vcharacteristic ~squares!. The data in ~b!
are fitted with a square-root dependence on
u
V12Vth
u
, where Vth is
defined in the text.
FIG. 5. ~a!Real-time traces as well as ~b!peak Ipand final
current I1compared with the I-Vcharacteristics ~solid line!for up
jumps starting from the third branch. Curves A,B, and Ccorrespond
to final voltages V150.798, 0.846, and 0.853 V, respectively. The
short-dashed line in ~b!indicates the separation between the mono-
pole and tripole relocation process, and the long-dashed line is used
as a guide to the eye.
ROGOZIA, TEITSWORTH, GRAHN, AND PLOOG PHYSICAL REVIEW B 65 205303
205303-4
the larger spike, a series of regular spikes ~region 3!appears
with a lower current level, before the current rises ~region 4!.
The traces are ensemble averages of about 100 measure-
ments in order to obtain a better signal-to-noise ratio, i.e., the
position of the individual spikes is essentially deterministic.
However, the delay time
t
d, defined in Fig. 5~a!, fluctuates
stochastically. Therefore, it is somewhat smeared out in the
ensemble-averaged traces of Fig. 6.
The separation of the regular spikes in region 3 is about
50 ns, which corresponds to a frequency of 20 MHz. This
frequency falls into the same range of frequencies for the
spikes present in current self-oscillations of 10–20 MHz,
which have been observed in the same sample for opposite
polarity.16–18
We have performed several sets of measurements to vary
the number of regular spikes in regions 3 and 4. For each set,
we select the starting voltage V0on a particular initial branch
(N0) and vary the final voltage V1to lie on different final
branches (N1) of the I-Vcurve. In these measurements, the
number of regular spikes Nrs in regions 3 and 4 can be esti-
mated by
Nrs'NSL2N02N1,~4!
as long as N01N1,NSL . The difference between the left-
and right-hand sides of Eq. ~4!varies typically between 1
and 3. For jumps with N01N1.NSL , regions 3 and 4 are
not observed.
According to the theoretical work by Amann et al.,9the
larger up jumps exhibit a more complex relocation scenario.
In order to understand this behavior, we have to consider the
electronic transport between the emitter contact and the first
SL barrier. The emitter is assumed to be Ohmic, with an
effective contact resistance
r
emitter.0. Its current-field char-
acteristic crosses the current-field characteristic for a homo-
geneous field distribution of the SL in the negative-
differential-resistance ~NDR!region at a certain critical
current level Icrit ~cf. Fig. 1 in Ref. 9!. Before the voltage
step is applied, a CDL is present between the emitter and the
first SL period according to Poisson’s equation, since the
field in the low-field domain is smaller than in the emitting
contact. This CDL contains a much lower carrier density
than the one at the domain boundary, since the field change
between the emitter and LFD of the SL is much smaller than
that between the LFD and HFD. After applying the voltage
step, we have to distinguish the following cases. As long as
Ip,Icrit , this CDL persists. However, when Ipbecomes
larger than Icrit , the current in the emitter becomes larger
than that in the SL. For very short current peaks, the critical
value Icrit can be exceeded without producing a traveling
CDL, since there is not sufficient time to convert the CDL
into a CAL. However, for sufficiently long current peaks, the
CDL between the emitter and LFD can be transformed into a
CAL. At the same time, a CDL is formed between the first
and second barriers of the SL, which will immediately begin
to move into the SL. As the CDL moves into the SL, the new
CAL between the emitter and the first barrier will also start
to move into the SL. In this case, the relocation process
involves two CAL’s and one CDL, which we will refer to as
atripole.
Figures 7~a!and 7~b!show the electron densities as a
function of time and space for the simple monopole and
tripole relocation processes, respectively. White and black
areas depict CAL’s and CDL’s, respectively. The four regions
of the tripole relocation process defined in Fig. 6 are also
indicated. The simple monopole relocation displayed in Fig.
7~a!refers to a small up jump. The larger up jumps contain
four regions, which correspond to four different phases of the
tripole relocation process. Phase 1 occurs during the initial
displacement current peak. During this time, the CAL be-
tween the LFD and HFD moves upstream toward its final
position. At the same time, the CDL at the emitter begins to
move as described above, leaving a HFD behind, which
grows with increasing time @cf. region 1 in Fig. 7~b!#. Since
the number of periods in the HFD increases, while the ap-
plied voltage remains constant, the effective field strengths in
FIG. 6. Ensemble-averaged current transients for switches from
the third to the fifth branch ~A!and from the tenth to the thirteenth
branch ~B!of the I-Vcharacteristic. Curve Bis shifted by a 25-
m
A
offset for clarity.
FIG. 7. Schematic evolution of the electron densities in the
quantum wells during the switching process via ~a!the simple
monopole and ~b!the tripole relocation process for a jump from the
eighth to the ninth branch. The CAL’s ~CDL’s!are indicated by
white ~black!areas. The emitter ~collector!is located near well 1
~40!. The numbers 1, 2, 3, and 4 on the right-hand side in ~b!refer
to the time intervals labeled in the same way in Fig. 6.
RELOCATION DYNAMICS OF DOMAIN BOUNDARIES IN . . . PHYSICAL REVIEW B 65 205303
205303-5
Figure 6.7: (a) Experimental current traces (upper panel) for up jumps from the
third branch to the third (A) and fourth branch (B and C), respectively.
The lower panel shows the final current (I1) and the maximum current
(Ip) during the jump. (b) Experimental current traces for switches from
the third to the fifth branch (A) and from the tenth to the thirteenth
branch (B) of the current voltage characteristic. The regions 2 and 3
correspond to the tripole and dipole phases, respectively. Reprinted
from [110].
to the new domain boundary. This is apparently in violation of the basic rules of
single front dynamics (see Chapter 3), which for instance require, that fronts of
same polarity move in the same direction. However in Chapter 3, we were dealing
with fully developed fronts, while the fronts appearing in Fig. 6.8(b) are only partly
developed. Although the monopole mechanism is possible theoretically, the plateau
like experimental current traces observed in Refs. [109, 110, 108] (cf. Fig. 6.9) show
that the dipole mechanism are more common experimentally.
88
6.1 Switching
+0.4V
+0.5V
+0.9V
time [µs]
0 10
-2 0 2 4 68 10
time [µs]
0
50
100
150
200
250
current J [µΑ]
+0.1 V
+0.4 V
+0.5 V
+0.9 V
Ustep
Figure 6.8: Current responses (left) and density evolution (right) for switching pro-
cesses for a superlattice as in Fig. 6.2, but with σ= 0.1 Ω−1m−1. Initial
voltage Ui= 1.5 V and various Ustep. The current traces in the left panel
are shifted vertically by multiples of 25µA.
Figure 6.9: Experimental current trace of an N= 20 well superlattice for a voltage
step from 0 to the second branch. Reprinted from [108].
89
6 Nonstationary External Voltage
6.2 Ramping
Experimentally it is not possible to switch the voltage instantaneously as in Eq. (6.1.1),
but the effectively applied voltage will rather follow the ramp like form:
U(t) =
Uifor t < −τr,
Ui+τr−t
τr(Ustep) for −τr≤t < 0,
Uf=Ui+Ustep for t≥0,
(6.2.1)
with the ramping time τr.
Let us first consider the situation, where the final voltage Ufis close to the
discontinuity point Udisc at which the current branch terminates. This situation
was studied experimentally in Ref. [47]. We see from Fig. 6.11 that the current
response depends sensitively on the exact value of the final voltage Uf. The branch
at which the operating point at Uiis located if Fig. 6.11 has its discontinuity at
Udisc ≈1.5738 V. Thus one could suppose that for Uf< Udisc the final operating
point is still on the original branch. This is however not the case, since for Uf=
1.5737V, the final operating point already is located on the next branch (cyan line
in Fig. 6.11). Here the current rises during the ramping time, then at t= 0 quickly
relaxes within less than 0.05 µs to a current which corresponds to an operating point
on the original branch. At this level the current remains almost constant for a delay
time of about τd= 1 µs, before it finally drops to its final operating point on the
next branch. The switching time for this final drop is about τs= 0.2µs. If we now
further decrease Uf, the switching time τsremains approximately constant, but the
delay time τdincreases dramatically (right panel of Fig. 6.11), and diverges shortly
before the final operating point remains on the same branch (green line in left panel
of Fig. 6.11). This agrees with the experimental data from Ref. [47] (cf. Fig. 6.10).
It is now also interesting to consider the distribution of the relocation times for
many switching processes with the same Uf. Experimentally it is not possible to
specify Ufwith arbitrary high accuracy and also the electron densities nmwill vary
slightly with each switching process. For Uffar away from Udisc, the delay time
τdcan be neglected, and we obtain τrel ≈τs≈0.2µs. Assuming a Gaussian dis-
tribution of Ufwe expect that the distribution of τrel will also be Gaussian with
a rather small variation σ. This is also found experimentally (cf. right inset in
Fig. 6.10). However from Fig. 6.11 we see that for Ufclose to the discontinuity
point Udisc, the relocation time is extremely sensitive to variations δUfof Uf. Addi-
tionally the sign of δUfis important. Consider for instance the case Uf= 1.57366V
(violet line in Fig. 6.11(b)) with a relocation time τrel ≈2.5µs. A variation of
Ufby δUf= 0.01 mV will yield a slightly lower τrel ≈2.0µs. But a variation by
δUf=−0.01 mV (red line in Fig. 6.11(b)) approximately triples the relocation time
to τrel ≈7.9µs. Thus we expect that a Gaussian distribution in Uftranslates to an
asymmetric distribution of the relocation time τrel with a pronounced tail at large
τrel. This behavior has also been observed experimentally, as shown in the left panel
90
6.3 Sweeping
wells.12–14 In this work we will focus on the switching times
between the third and fourth current branch.
Typical time traces for switching from a fixed initial volt-
age on the third branch (V05 2 709 mV) to several final
voltages V1on the fourth branch are shown in Fig. 2. V0and
the V1values labeled Ato Fare marked in the I-Vcharac-
teristics in the inset. After switching the voltage, the current
increases immediately to a value corresponding to the un-
stable part of the third branch, i.e., to a value, that can be
reached by linear continuation of the third branch beyond the
current jump. For large voltage jumps ~cf. Fin Fig. 2!, the
current reaches its final value on the fourth branch on a time
scale of a few hundred nanoseconds. If V1is reduced ap-
proaching the voltage of the current jump for the up sweep,
the current is almost constant ~with a slight decrease!over a
delay time
t
d, until it changes rapidly with a switching time
t
s. In fact, when the current value during
t
dfalls below a
critical level, the current switches to its final value. The
switching time increases by a factor of 3.5 going from Fto
A. However,
t
dincreases from 200 ns for large values of V1
up to more than 20
m
s for V1just above the current jump of
the up sweep (Fto Ain Fig. 2!.
In order to analyze the delay times in terms of their sta-
tistics, we have directly measured the distribution functions
using the built-in functions of the oscilloscope. Two ex-
amples are shown as insets in Fig. 3. There is a very pro-
nounced change in the shape of the distribution function go-
ing from larger to smaller values of
u
V1
u
. The main part of
Fig. 3 shows the averaged response times
t
¯
~dots!and the
respective widths
s
~open squares!of the distributions on a
logarithmic scale as a function of V1. Both strongly increase
with decreasing
u
V1
u
(Fto Ain Fig. 2!. For values of V1far
away from the current jump, the distribution function be-
comes very narrow ~cf. inset for V15 2 752.5 mV in Fig. 3!
and exhibits a symmetric, Gaussian-like shape. However, for
values of V1close to the current jump, the distribution func-
tion has a completely different, asymmetric shape with a
steep increase at shorter times and a broad tail at longer
times ~cf. inset for V15 2 732.0 mV in Fig. 3!. Note that
also the time scale has changed by more than one order of
magnitude. The fits through the data points of the distribu-
tion functions will be discussed later.
In previous experiments,3,4,6 the bias was changed from 0
V to its final value in order to study the formation of the
domains, i.e., the boundary is moving over many SL periods.
Here, we investigate the relocation of the domain boundary
over a single period so that the domains are already formed
by applying a finite bias V0. However, in contrast to Ref. 5,
we are now interested in the switching time distribution and
its dependence on the final voltage V1. Our results can be
interpreted in the following way. When V1is located on the
FIG. 1. Time-averaged I-Vcharacteristics of the first five
branches ~1, 2, 3, 4, 5!for up sweep and down sweep between 0 and
26 V as indicated by the arrows at 5 K. The full I-Vcharacteris-
tics for both sweep directions is shown in the inset.
FIG. 2. Typical time traces for switching from V0on branch 3 to
different voltages (Ato F) on branch 4. The switching time
t
is
indicated for case B. The closer the final voltage is to the beginning
of the fourth branch, the longer the switching time. The inset dis-
plays the time-averaged I-Vcharacteristics indicating V0and the
different final voltages V1labeled Ato F.
FIG. 3. Average value
t
¯
~dots!and width
s
~open squares!of
the distribution of the relocation times for switching from V05
2650 mV on branch 3 to different final voltages on the branch 4.
The insets show two examples of the distribution functions mea-
sured at a voltage of V15 2 732 and 2752.5 mV.
RAPID COMMUNICATIONS
ROGOZIA, TEITSWORTH, GRAHN, AND PLOOG PHYSICAL REVIEW B 64 041308~R!
041308-2
Figure 6.10: Average (¯τ) and width (σ) of the experimental relocation time distri-
butions (insets) for up jumps from the third to the fourth branch at
different final voltages. The discontinuity of the third branch is close
to V=−732.0 mV. Reprinted from [47].
of Fig. 6.10. In the literature, however there exist two alternative explanations for
this distribution, which propose that either thermal noise [116] or single electron
tunneling [112] are responsible for the uncertainty in the relocation time. Such
effects would however only dominate, if the final voltage Ufcould be experimen-
tally fixed with a very high precision, and the variation of Ufas the source of the
distribution of the relocation time seems to be the more natural explanation.
6.3 Sweeping
Instead of only ramping to the next branch discontinuity, it is also interesting to
sweep the voltage over several branches. We again use a voltage profile of the form
(6.2.1).
A typical result of the current voltage characteristic for different sweep velocities is
shown in Fig. 6.12(a). A large ramping time, such as τr= 10 µs, merely reproduces
the up sweep branch of the stationary current voltage characteristic in Fig. 6.2.
With decreasing τr, the sharp drops at the discontinuity points of the branches are
smeared out, and at the same time the current level rises (cf. τr= 4,2,0.7µs).
For even smaller τr= 0.6µs (orange line in Fig. 6.12), the current shows a
fundamentally different behavior. It first rises to a peak value of about 250 µA at
U= 1.5 V, and then drops to current values about 120 µA, with oscillations of about
15 µA. The explanation for this phenomena is given again by the corresponding
electron density plots (Fig. 6.12(b)). While for τr= 0.7µs the original accumulation
91
6 Nonstationary External Voltage
1.57 1.575 1.58
voltage U [V]
40
60
80
100
120
140
current J [µΑ]
0.01
0.1
1
10
100
τd[µs]
-2 0 2 4 6 8
time [µs]
100
150
200
250
300 Uf
1.573649
1.573650
1.573651
1.57366
1.5737
1.574
1.58
a) b)
Figure 6.11: Ramping processes with Ui= 1.5 V and various final voltages Uf(col-
ored points in (a)) close to the discontinuity of the original branch.
The ramping time is τr= 100 ns. The current traces in (b) are shifted
vertically by multiples of 25 µA.
front moves towards the emitter, for τr= 0.6µs, a new dipole is injected at the
emitter, which is triggered by the large current peak. The leading depletion front
of the dipole eventually merges with the original accumulation front, similar to the
dynamics found for large positive switching voltages (cf. Fig. 6.6). However since
the external voltage still increases during the tripole phase, the front trajectories in
Fig. 6.12(b) are distorted, and the current during the tripole phase is larger than
the classical tripole current j(2,1).
The findings of this section agrees well with experimental results [108, 110]. In
particular in Ref. [110] (cf. Fig. 6.13) the importance of the triggering current was
explicitly demonstrated.
6.4 Impedance
We now consider the response of a superlattice to a time dependent external voltage
of the form
U(t) = Udc +Uac sin (2πt/τac),(6.4.1)
which is the sum of a dc voltage part Udc, and an ac part which is characterized
by the amplitude Uac and the frequency ωac = 2π/τac. Let us assume that for
Uac = 0, the superlattice contains a moving dipole consisting of an accumulation
and a depletion front moving to the collector. From Sec. 3.2 we know that the
fixed external voltage causes both fronts to move with the same velocity (va=vd),
which in turn fixes the current to jd(cf. Fig. 3.13). This is of course not true, if
the external voltage is not fixed. For example, if the external voltage increases the
high field domain should grow, which is only possible, if the depletion front moves
92
6.4 Impedance
time [µs]
0 10
0.6
0.7
τr [µs]
11.5 22.5 3
voltage [V]
100
150
200
250
300
current J [µΑ]
0.6 µs
0.7 µs
2 µs
4 µs
10 µs
b)a)
Figure 6.12: Current voltage characteristics (a) and density plots (b) for sweeping
from Ui= 1 V to Ui= 3 V with various τr.
the LFD and HFD decrease so that the current is reduced
according to the homogeneous current-field characteristic.
Phase 2 begins after
t
p, when the current has dropped
below Il. At this point in time, there are three traveling
layers separating the field in the SL into two LFD’s and two
HFD’s @cf. region 2 in Fig. 7~b!#. Both CAL’s move with the
same velocity toward the collector ~cf. Ref. 9!. Because of
the constant total voltage, the number of periods in the
HFD’s must now remain constant. Therefore, the sum of the
velocities of the two CAL’s has to be the same as the velocity
of the CDL, i.e., 2 vaccu5vdepl . Since the average current
should have a value for which the CDL has twice the veloc-
ity of the CAL’s,9a rather low current value is observed in
Fig. 6. The jumps of the two CAL’s across individual quan-
tum wells appear as irregular spikes within region 2 in Fig. 6,
so that they seem to be uncorrelated.
After the original CAL reaches the collector, which is
indicated by a larger spike in Fig. 6, the tripole reduces to a
dipole and phase 3 begins @cf. region 3 in Fig. 7~b!#. The
velocities of the CAL and CDL are now the same. Since the
CDL is extended over several periods, the current transients
are dominated by the motion of the CAL, which now appears
as regular spikes in Fig. 6. When the CDL reaches the col-
lector ~after the time
t
dipole!, phase 3 is completed.
In phase 4, only a CAL is present in the SL. Since now
the number of periods in the HFD decreases with increasing
time, the field strengths of LFD and HFD have to increase,
resulting in an increase of the current. The CAL continues to
move toward its final position. After reaching the well adja-
cent to its final position, the situation becomes exactly the
same as for the down jumps discussed above. The current
remains for a stochastically varying delay time
t
din this
well, where
t
ddepends in the same way on the voltage to the
current discontinuity, as discussed in Ref. 7. Therefore, the
described behavior also explains the observation in Fig. 5
that even for larger up jumps the final current level is located
on the down-sweep characteristic.
We would like to point out that this complex relocation
scenario for larger up jumps may occur only in the first pla-
teau of the I-Vcharacteristic. The current-field characteristic
of the emitter contact may cross the NDR region of the first
resonance, but not of any higher resonance. Under this con-
dition, the tripole relocation process is only observed for the
first plateau.
VI. MEASUREMENTS WITH DIFFERENT TYPES
OF VOLTAGE STEPS
For fast voltage steps (Dt'8 ns), the displacement cur-
rent peak can be well above Icrit . When the step is replaced
by a ramp with a certain length Dt, the displacement current
peak will be reduced according to Eq. ~2!. For very long
ramping times, it will fall below Icrit , so that the CDL cannot
be formed. Figure 8 shows the current transients for jumps
from the third to the fourth branch for Dtbeing varied be-
tween 10 and 120 ns. For short ramping times, Ipis large,
resulting in the formation of the CDL. With increasing ramp-
ing time, Ipdecreases. For Dt560 ns, the formation of the
CDL seems to be delayed. In this case, Ip5110
m
A, which
is still above Icrit . For an even longer ramping time (Dt
5120 ns), the current transient changes to the type observed
for simple monopole switching so that Iphas to be smaller
than Icrit . Therefore, we can estimate Icrit to be about
105
m
A.
In order to obtain even more information about the value
of Icrit , we performed triangular voltage sweeps with four
different sweep rates as shown in Fig. 9. For a sweep rate of
10 mV/
m
s and below, the I-Vcharacteristic becomes very
similar to the one for dc measurements ~thin lines in Fig. 9!.
However, with increasing sweep rate, the jumps to the next
branches occur at higher ~lower!voltages for the up ~down!
sweep so that these jumps take place at higher ~lower!cur-
rent levels. This observation implies that the monopole at the
domain boundary cannot follow the voltage sweep anymore,
because it needs a certain time to jump from one well to the
adjacent well. In the depicted range of sweep rates, the dis-
placement current Idisp is below 1
m
A and can therefore be
neglected.
At a sweep rate of 300 mV/
m
s@cf. Fig. 9~b!#, the current
of the 11th branch reaches Icrit , triggering the formation of a
CDL. This can be seen from the strongly reduced current in
comparison to the case with lower sweep rates. When the
sweep rate is increased further, Icrit can already be reached on
the eighth branch. This observation is an indication that Icrit
does not depend on the actual current maximum of each
FIG. 8. Ensemble-averaged current traces for sweeps from the
third to the fourth branch with different ramp times depicted in the
inset. Below a certain peak current ~horizontal dashed line!, the
relocation process changes from a tripole process to an ordinary
monopole relocation process.
FIG. 9. Current vs voltage for triangular voltage sweeps from
the fifth to the twelfth branch for different sweep rates ~a!280
mV/
m
s, ~b!300 mV/
m
s, and ~c!350 mV/
m
s in comparison with a
low sweep rate of 10 mV/
m
s. If the critical current (Icrit
5105
m
A) is reached, the tripole process begins.
ROGOZIA, TEITSWORTH, GRAHN, AND PLOOG PHYSICAL REVIEW B 65 205303
205303-6
Figure 6.13: Experimental current voltage characteristic for triangular voltage
sweeps at different sweep rates. The broken line denotes the critical
current which triggers the dipole generation at the emitter. Reprinted
from [110].
93
6 Nonstationary External Voltage
faster than the accumulation front. More precisely, from (4.2.1) we get
vd(j)−va(j) = −˙
U
Fh+U∂jFh(j)
(Fh)2
∂j
∂t ,(6.4.2)
where we have used the approximation Fl≈0. Similarly to (4.2.9) we approximate
the left hand side of (6.4.2)by
vd(j)−va(j)≈kv(j−jd),(6.4.3)
kv= (∂jvd(jd)−∂jva(jd)) ≈2d
eND
.(6.4.4)
For small voltages, we can neglect the second term on the right hand side of (6.4.2),
and simply get
Fhkv(j−jd) = −˙
U=−iωU. (6.4.5)
This gives rise to a purely imaginary impedance of
Z=U
A(j−jd)=iFhkv
ωA .(6.4.6)
For a superlattice of type B we have Fh≈5 MVm and kv≈1.6·10−4m3/Cb.
For τac = 1ns, this yields,
Z=i·130 nΩ.(6.4.7)
From the numerical simulation in Fig. 6.14, we obtain a somewhat smaller value
Znum ≈i60 nΩ, but we see that the expected phase relation, between current
density and the external voltage is accurately fulfilled.
We may note that (6.4.6) is only valid in the high frequency limit, when τdc
is much smaller than the typical lifetime of the fronts. Furthermore the fronts
have to be well separated from each other, in order to fulfill the velocity current
characteristic for single fronts (cf. Fig. 3.13). Since this last condition is not fulfilled
for the oscillation mode considered in Refs. [49, 117], their results can not be directly
compared to the results in the present section.
94
6.4 Impedance
190 192 194 196 198
time [ns]
-2.5
-2
-1.5
-1
current density [A/mm2]
190 192 194 196 198
0.55
0.54
0.53
0.52
0.51
0.5
0.49
0.48
voltage [V]
Figure 6.14: Current density trace (blue) for superlattice of type B with σ=
1.3 Ω−1m−1,Udc = 0.5 V Uac = 0.01 V and τac = 1 ns. The cyan
line shows the time averaged current density over 0.05 ns. The electron
density evolution is similar to Fig. 3.1(b).
95
6 Nonstationary External Voltage
96
7 Front Dynamics in Two Spatial
Dimensions
Up to now we have considered the superlattice as a one dimensional device, and
assumed that at any time each quantum well is homogeneously charged. In that case
only the vertical charge transport from one quantum well to the next is responsible
for the observed dynamical patterns. Such an assumption is only justified for small
lateral well sizes, since then the relaxation time for charge fluctuations within one
well is much faster than all other dynamical time scales.
In this chapter we will consider superlattices with large lateral extension. Then
lateral patterns may contribute to the overall dynamics of the system. In particular
the interaction between vertical and lateral patterns may give rise to qualitatively
new scenarios.
7.1 Lateral Transport Theory
7.1.1 Dynamical Equations
Taking into account the lateral degrees of freedom, the new dynamical variables
are the two dimensional charge densities nm(x, y), which now, in addition to the
quantum well index m, also depend on the well plane coordinates x∈[0, Lx] and
y∈[0, Ly] (here Lxand Lyare the extensions of the superlattice in xand ydirection,
respectively). The continuity equation (3.0.1) then generalizes to the new dynamical
equation
e˙nm(x, y) = jk
m−1→m−jk
m→m+1 −∇⊥j⊥
mfor m= 1, . . . N, (7.1.1)
with
jk
m→m+1(x, y) = jm→m+1(Fk
m, nm, nm+1) (7.1.2)
∇⊥=ex
∂
∂x +ey
∂
∂y ,(7.1.3)
dj⊥
m(x, y) = −eµnmF⊥
m−eD0∇⊥nm.(7.1.4)
Here the lateral two–dimensional current density j⊥
m(units: [A/m]) is the sum of
a drift and a diffusion term, characterized by the mobility µand the diffusion
97
7 Front Dynamics in Two Spatial Dimensions
coefficient D0, respectively. The electric fields fulfill the semi-discrete version of
Gauss’s law
Fk
m−Fk
m−1+d∇⊥F⊥
m=e
r0
(nm−ND) for m= 1, . . . N, (7.1.5)
with the boundary conditions
U=−
N
X
m=0
Fk
m(x, y)dfor x∈[0, Lx], y ∈[0, Ly],(7.1.6)
F⊥
m(0, y) = F⊥
m(Lx, y) = F⊥
m(x, 0) = F⊥
m(x, Ly) = 0.(7.1.7)
7.1.2 The Generalized Einstein Relation
The parameters µand D0appearing in (7.1.4) are the mobility and the diffusion
constant within the well, respectively. They are connected by a generalized form of
the Einstein relation [118, 119]
D0(nm) = nm
−eρ0(1 −exp [−nm/(ρ0kBT)])µ, (7.1.8)
with ρ0=m/(π~2). Note that the mobility µcan in principle also depend on n,
and that (7.1.8) can only be derived for the equilibrium case. In the following we
make the assumptions that µis fixed (for GaAs we assume µ≈10 m2/Vs), and
that (7.1.8) is still valid in the non-equilibrium case. Then we may rewrite (7.1.4)
as
j⊥
m(x, y) = −eµnmF⊥
m+∇⊥nm
−eρ0(1 −exp [−n/(ρ0kBT)]).(7.1.9)
7.1.3 Solving Poisson’s Equation
For the integration of (7.1.1), it is necessary to solve (7.1.5) efficiently with respect
to the electric fields F⊥
mand Fk
m. This amounts to the solution of the semi-discrete
Poisson equation for the potential ϕm(x, y) of the form
∆ϕm(x, y) = ∆⊥+ ∆kϕm(x, y) = −e
dr0
(nm−ND) for m= 1, . . . N, (7.1.10)
with
∆⊥ϕm(x, y) = ∂2
∂x2+∂2
∂y2ϕm(x, y),(7.1.11)
∆kϕm(x, y) = ϕm−1(x, y)−2ϕm(x, y) + ϕm+1(x, y)
d2.(7.1.12)
A straightforward way for obtaining the potential ϕmfrom (7.1.10) would be to
calculate the capacity matrix ∆−1explicitly. With Mthe number of discretization
98
7.1 Lateral Transport Theory
points in the (x, y) plane, this matrix would however have (NM)2elements, and we
have to perform O(N2M2) operations at every integration time step.
In search of a more efficient algorithm, we compare the contributions from ∆⊥ϕm
and ∆kϕmin (7.1.10). In Ref. [119] the mean free path of the degenerate electrons in
the well was estimated as lm≈0.3µm, and we may expect that typical structures in
the lateral direction vary on a length scale which is even larger. Indeed it was found
by numerical simulation of the very similar DBRT model that lateral structures
typically occur on the length scale of about lt= 10 µm [21, 120, 121]. On the other
hand, the variations of the potential in the vertical direction zis of the order of the
superlattice period d≈10 nm. We may therefore conclude that
∆⊥ϕm∼(lt)−2∆kϕm∼d−2.(7.1.13)
This allows to invert the Laplace operator by the use of a perturbation expansion
of the form,
∆−1= (∆⊥+ ∆k)−1= (1 + ∆−1
k∆⊥)−1∆−1
k,(7.1.14)
≈(1 −∆−1
k∆⊥+...)∆−1
k= ∆−1
k−∆−2
k∆⊥+. . . . (7.1.15)
In the last step we used the fact that ∆⊥and ∆kcommute. Applying (7.1.14) to
(7.1.10) then yields
ϕm(x, y) = ϕ0
m(x, y) + ϕ1
m(x, y) + . . . , (7.1.16)
ϕ0
m(x, y) = −e
dr0
∆−1
k(nm−ND),(7.1.17)
ϕ1
m(x, y) = + e
dr0
∆−2
k∆⊥nm.(7.1.18)
The advantage of such a solution for (7.1.10) lies in the fact that it can be calculated
very efficiently. ϕ0
m(x, y) is evaluated by shooting with
˜ϕ0
0(x, y) = ˜ϕ0
1(x, y) = 0,(7.1.19)
˜ϕ0
m+1(x, y) = 2 ˜ϕ0
m−˜ϕ0
m−1−ed
r0
(nm−ND) for m= 1 . . . N, (7.1.20)
and then taking into account the corrections from the boundary conditions by
ϕ0
m(x, y) = ˜ϕ0
m+ (U−˜ϕ0
N+1)m
N+ 1 for m= 1 . . . N + 1.(7.1.21)
The algorithm described by (7.1.20) and (7.1.21) completes in only O(NM) oper-
ations.
Also the matrix multiplication ∆⊥nmappearing in the calculation of ϕ1
m(x, y) in
(7.1.17) is of O(NM), since ∆⊥is a matrix with only five entries per row. The
operator ∆−2
kis simply evaluated by applying the algorithm of (7.1.20) twice and
use a correction as in (7.1.21), but with U= 0.
99
7 Front Dynamics in Two Spatial Dimensions
0.9 0.95 11.05 1.1
Voltage [V]
4
5
6
7
8
9
10
current density [A/mm2]
well #88
well #90
well #89
Figure 7.1: Homogeneous current voltage characteristic for superlattice of type B,
with σ= 500 Ω−1m−1and Lx= 50 µm. The green lines denote the
voltages considered in Fig. 7.3.
Once we have obtained the potential ϕm(x, y), the electric fields are easily ob-
tained in O(NM) by
Fk
m(x, y) = ϕm+1(x, y)−ϕm(x, y)
d,(7.1.22)
F⊥
m(x, y) = −∇⊥ϕm(x, y),(7.1.23)
and can be used in (7.1.9) and (7.1.2) to calculate the current densities for the
electron density evolution equation (7.1.1).
7.2 Stability of Inhomogeneous Lateral Patterns
For the numerical implementation of the scheme described in the Sec. 7.1, we use
a superlattice of type B (see Table 2.2 on page 12), with a contact conductivity
of σ= 500 Ω−1m−1. Here we choose a large σ, in order to avoid front generation
processes at the emitter. For simplicity we assume that the sample extension in the y
direction is small, such that pattern formation can only occur in the xdirection. We
choose Lx= 50 µm and M= 25 discretization points. We calculate ϕm(x, y) only
to the lowest order, and assume an effective diffusion constant of D0≈0.01 m2/s.
In the homogeneous case without lateral pattern formation, the superlattice shows
a stationary current voltage characteristic with branches associated with the peak of
the electron charge distribution (Fig. 7.1) located in the well labeled by its number.
100
7.2 Stability of Inhomogeneous Lateral Patterns
00.5 11.5 22.5
time [ns]
4
5
6
7
j [A/mm2]
1.10 V
1.00 V
0.98 V
0.97 V
Figure 7.2: Current trace for superlattice parameters as in Fig. 7.1, with inhomo-
geneous initial condition at various voltages. At t= 0, accumulation
fronts are placed in the left half of well 90 and the right half of well 88.
Due to the multistability apparent from Fig. 7.1, we may expect stable lateral
patterns, where the superlattice is divided along the xaxis into regions with varying
operating points. We prepare initial conditions, with the left and right halves
of the superlattice corresponding to operating points on branch number 90 and
88 respectively. This is achieved by putting electron accumulation fronts at the
appropriate positions in well 90 and 88. We then study the response of this initial
configuration to various voltages.
The resulting current traces and electron densities are shown in Fig. 7.2 and
Fig. 7.3, respectively. We see that for U= 0.97 V the sharp current peak from
the switch on of the external voltage, pushes the accumulation front in the left
half from well 90 to 89 already at t= 0.1 ns. The current density is then given
by the average of the current densities of the operating points at well 88 and 89.
Subsequently the accumulation front at well 89 extends to the right and extrudes the
accumulation front at 88, until at t= 3.5 ns we arrive at a homogeneous state with
operating point at well 89. During this process the current density rises linearly
to the value of the final operating point. For U= 0.98 V, we observe a similar
behavior, but the time until the final operating point on branch 89 is reached, has
approximately doubled. Note that the branch 89 wins over branch 88, although
according to Fig. 7.1, both branches are stable at this voltage. This changes for
U= 1.0V, however, where the operating points on well 89 and 88 coexist (Fig. 7.3)
for longer than the simulation time, and the final current is given by the average of
the currents from both branches.
For an even higher voltage U= 1.1 V, we find that the switch-on-peak shifts both
accumulation fronts by one well from well 90 and 88 to 89 and 87, respectively,
within less than 0.1 ns. Then the electron accumulation from well 89 drops to well
88 starting from the middle of the sample, until at t= 7 ns we reach a stable
101
7 Front Dynamics in Two Spatial Dimensions
1.00
1.10
0.97
U [V]
0 ns 0.1 ns 2 ns 3 ns
0 ns 0.1 ns 7 ns 9 ns
0 ns 1 ns 6 ns 8 ns
x
0.98
1 ns 2 ns 4 ns 6 ns
z (well)
Figure 7.3: Electron density evolution for inhomogeneous initial conditions as in
Fig. 7.2, shown in the (x, z) plane of the superlattice.
102
7.2 Stability of Inhomogeneous Lateral Patterns
0 ns 0.2 ns 0.4 ns 0.6 ns
0.1 ns 0.12 ns 0.14 ns 0.16 ns
0.8 ns
0.26 ns
well # well #
Figure 7.4: Superlattice as in Fig. 7.1 at U= 1.0V. Inhomogeneous initial condition
with accumulation (depletion) front in left (right) half of well 80.
configuration with the left (right) half of the sample on branch 88 (87). During
this transition, the current density drops linearly, as expected from the weighted
average of the three involved operating points.
It is also interesting to consider an initial condition, with an accumulation front
in the left half, and a depletion front in the right half of the same well. Such
a configuration is shown in Fig. 7.4. We see that a new accumulation front is
generated at the emitter in the right half of the sample, and moves towards the
collector. Together with the already present fronts, we thus obtain a dipole in the
left, and a monopole in the right part of the sample. As the dipole moves towards
the collector, the monopole extends towards the right, until it eventually occupies
the whole sample width. This behavior is also reflected by the corresponding current
trace (cf. Fig. 7.5), which for t > 1.0 ns can be explained as the weighted average
of the dipole current jdand the current of the final operating point.
In summary we have seen in this chapter that the lateral structures in super-
lattices reveal new aspects of front interaction processes, which are fundamentally
different from the purely one–dimensional vertical interaction scenarios of Chap-
ter 3. However, a more systematic analysis is still necessary, to gain a thorough
understanding of the involved mechanisms. Such an analysis is however beyond the
scope of the present work.
103
7 Front Dynamics in Two Spatial Dimensions
00.5 11.5 22.5
time [ns]
2
3
4
5
6
7
j [A/mm2]
Figure 7.5: Current trace for the scenario in Fig. 7.4
104
8 Summary and Outlook
Semiconductor superlattices have been a focus of intensive research during the last
decade. Experimentally a host of intriguing phenomena, such as self sustained high
frequency oscillations, stationary field domains, or a remarkable response to the
change of the external voltage has been observed. Theoretically, the dynamics of
the superlattice has been described by various theories on different hierarchy levels.
Examples are the Wannier-Stark ladder, the miniband transport model or the non-
equilibrium Green’s function theory. One reason for this interest in superlattices can
be attributed to the expected technological applicability, for instance in Terahertz
electronics, but from a more fundamental point of view, the significance of the
superlattice as a nonlinear model system, is equally important. In particular the
fact that it provides a rich front dynamics, has proved to be fruitful throughout this
work.
We founded our analysis on a semiclassical sequential tunneling model for the
electrons, which is motivated by quantum mechanical considerations. The resulting
nonlinear transport equations give rise to the formation of electron accumulation
and depletion fronts, which form the boundaries between high and low field do-
mains. It is thus natural to look for a description of the superlattice dynamics in
terms of fronts. Such a front model provides a new hierarchical level on top of the
semiclassical model.
With this aim, we first study the propagation, generation and annihilation of
single fronts in detail. It was found in Chapter 3 that the front velocities are deter-
mined by the overall current density, while the generation of fronts at the emitter
is governed by the nature of the emitter contact, characterized by the contact con-
ductivity σ. Fronts disappear from the system, as they either reach the collector, or
collide with a front of opposite polarity. It is this latter possibility of front annihila-
tion that allows for particularly interesting scenarios, such as chaotic behavior under
fixed external voltage conditions. We have demonstrated in Chapter 4 that large
parts of the bifurcation scenarios of the microscopic model can be reproduced by a
model which uses the front positions as the dynamical variables. This front model
has a very generic structure, and it may be relevant for other globally constrained
front systems as well.
As shown in Chapter 5, a further simplification of the front model applies, if
the fronts do not reach the collector. In this case the front model maps to a tank
model, which describes the filling heights of a number of water tanks. The tanks are
filled and drained by a given set of rules. Similar models are obtained generically in
various areas of science and engineering, for instance in the context of production
105
8 Summary and Outlook
processes. The tank model can be described analytically by iterated maps. In
Chapter 5 we explicitly constructed the maps for the first two nontrivial cases n= 3
and n= 4. It is shown that the map for n= 3 follows the universal U-sequence
of periods, which appears in a large class of one dimensional iterated maps. This
finally explains the peculiar bifurcation scenario observed in the microscopic model.
Experimentally, the current response of superlattices shows surprising features,
under sweeping or switching of the external voltage. For example, it is possible to
reach the down sweep branch of the stationary current voltage characteristic by a
fast voltage increase. As demonstrated in Chapter 6, such effects can be naturally
explained in terms of the front dynamics.
Apart from the fronts in vertical direction, fronts are also possible in lateral di-
rection. In Chapter 7 we extend the superlattice to include one additional lateral
dimension, and found that the stability of the corresponding lateral patterns de-
pends sensitively on the applied voltage.
Although the front model explains a large part of the superlattice dynamics, there
are nevertheless scenarios, which are not yet reproduced successfully, and should
be the object of future research on this subject. One failure of the front model
obviously arises from fronts, which are not fully developed. Here the model could
possibly be extended to at least take into account the “excitonic” fronts, which
were discussed in Sec. 4.1.1, and appear to be the most important correction to the
front model. It would also be worthwhile to understand, which parameters of the
superlattice influence the success of the front model. It may then be possible to
find a clear n= 4 or n= 5 scenario as in Fig. 4.11, in the microscopic model. Such
considerations may also help the experimentalists to construct superlattices, which
show self generated chaoticity. Furthermore the front model does not satisfactorily
reproduce the case where fronts reach the collector, and a more detailed analysis of
the collector contact seems to be necessary.
Interesting new effects are also expected from a further analysis of the lateral
instabilities in the superlattice. In this work we have only laid out the basic lateral
transport theory, but a detailed understanding of the interaction between vertical
and lateral fronts is still missing.
Last but not least a more ambitious extension of this work would be to check the
universality of the presented methods by applying them in a more general context.
As a first step it would be useful to derive a generalized front model from a more
generic, probably continuous model. A second step would then be to find concrete
examples of front systems, which can be described by such a front model. The
hope is, that in the course of this work, a unified theory encompassing general front
interactions will emerge.
106
Acknowledgement
I am grateful to my supervisor Prof. Eckehard Sch¨oll for his valuable advice and
support. He directed me to interesting topics of research and at the same time
permitted me to follow my own ideas. I am indebted to Dr. Andreas Wacker for
introducing me to the details of the superlattice model, and for many helpful hints
throughout this work. I would like to thank Prof. Harald Engel for making the
second assessment of this thesis.
I appreciate the collaboration with the experimentalists Marco Rogozia in Berlin
and Dr. Ekkehard Schomburg in Regensburg who provided the experimental basis
for this theoretical research.
This work also profited from the invitation of Prof. Luis Bonilla to visit his group
in Madrid and the subsequent collaboration.
I thank Anne-Katharina Jappsen for her contributions in the non stationary volt-
age case, Dr. Toni Lee for the vivid discussions on decoherence, Karsten Peters for
the intense collaboration on the tank model, Dr. Pavel Rodin for his help on the
lateral dynamics and Jan Schlesner for his extensive numerical support which sub-
stantiated many theoretical concepts of this thesis.
Last but not least I want to thank the present and former colleagues at the ITP
for the enjoyable atmosphere, in particular Reinhard Wetzler, with whom I shared
an office for the past years.
This work was supported by DFG in the framework of Sfb 555.
107
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